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DO = u< OU 160208 7) ~X ^ DO JOSMANIA UNIVERSITY LIBRARY Call Na S S7 5 C 7$ M Accession No. Z Author ThisTjook should be returned on or before the Bate last manrcd below. METHODS IN CLIMATOLOGY LONDON: GEOFFREY CUMBERLEGE OXFORD UNIVERSITY PRESS METHODS IN CLIMATOLOGY BY VICTOR CONRAD HARVARD UNIVERSITY CAMBRIDGE, MASSACHUSETTS HARVARD UNIVERSITY PRESS 1946 COPYRIGHT, 1944 BY THE PRESIDENT AND FELLOWS OF HARVARD COLLEGE Second Printing Printed in the United States of America by Lancaster Press, Inc., Lancaster, Penna. TO IDA CONRAD MY COMRADE IN LIFE AND WORK THIS BOOK is DEDICATED PREFACE CLIMATE influences the surface of the earth, and this con- versely, in its varieties, determines the climate under other- wise equal conditions. This intimate mutual connection makes climatology and climatography appear as parts of geog- raphy, because they are essentially necessary to describe the sur- face of the earth and its changes. These ideas find their expression in the fact that generally in colleges and universities, climatology as a whole is treated in the geographical departments. Perhaps the dependent role of climatology may be attributable also to the fact that geographers have so greatly furthered this science. For the most part, climatographies have been written by ge- ographers. Therefore, geographical methods are kept in the fore- front, and specifically climatological methods are not so much used. It would be satisfying if this book offered a bridge connecting the two realms. To judge from the author's many years of experience in Europe, followed by a few years in the United States, the student in ge- ography has perhaps sufficient knowledge of climatology, but he does not know how to deal with the original data of observations. He is, I should say, familiar with the results but not with the ways of getting them. This holds, particularly, for the different methods of mathematical statistics which can be adapted to an individual representation of the climates; only thus can too-schematical descriptions be avoided. This book should make the student acquainted with a number of methods of mathematical statistics and theory of probability applicable at once to climatological problems. Approximations are used as much as possible in order to facilitate arithmetic. The student who is interested in the mathematical proofs and deriva- tions as well as in their philosophical background must be referred to the special technical literature. The general introduction presents climatology as a world science, and its international organization. The number of ob- servations in the meteorological register makes the necessity of statistical methods evident. Their special application to clima- tological series is discussed. An instruction, easy to understand, vu viii METHODS IN CLIMATOLOGY is given for computing periodical phenomena by means of harmonic analysis. In the foregoing, the overwhelming role of the frequency distri- bution has been emphasized. This idea led L. W. Pollak to intro- duce a new system of treating climatological observations by means of automatic statistical machines. This system covers the entire body of the observational data and may represent the future of rational climatological statistics. Pollak's method is not dis- cussed here, however, because only with the resources of great institutions is it practicable. The general representation of climatological elements, factors, etc., is followed by discussions of the special characteristics of the single elements. The radiation elements are not dealt with. As far as frequen- cies and different correlations with other elements are concerned, the methods are identical with those applied to other climatological elements, on the one hand. On the other, such a discussion would have had to include the physical and astronomical nature of the subject. This exceeds the range of this book. The first two parts of the preseat book are concerned with the variations of the elements in the course of time at one fixed place. The third part presents the comparison of the elements which are observed synchronously at different places, and arrives at their geographical distribution. Critical scrutiny of the observational data leads to the examina- tion of the homogeneity of climatological series and to the reduc- tion to an identical period. These problems remind me of Hann's quotation of Francis Bacon, the great English philosopher: Si quis hujusmodi rebus ut nimium exilibus et minutis vacare nolit, imperium in naturam nee obtinere nee regere poterit. 1 These efforts are of fundamental importance not only for the climatologist but also for everybody who is interested in clima- tological series, as are the long-range forecaster and those dealing with periodicities and tree-growth analysis. Comparisons of data at different places reveal the concept of coherent and non-coherent climatic regions and of the climatic divide. These ideas show also clearly that the theory of correla- tion in the realm of climatology should play a significant role. An intermediate chapter deals with this subject. 1 "A scholar who is unwilling to take pains over such investigations because they seem too insignificant and microscopic will be able neither to gain nor to maintain mastery over nature." PREFACE ix Climatological examples demonstrate the computations so that everyone with high-school training should be able to under- stand the mathematical procedure. Linear correlation, simplifica- tions of the calculation, and regression equations are discussed. Finally, the formulas of partial correlation of three variants are added. Rather much space is devoted to graphical methods of representation, especially in connection with mountainous regions. The great importance of anomalies and isanomals is shown and supported by examples. At the end of this chapter, air-mass climatology, continentality, etc., are mentioned. The fourth section gives suggestions for the arrangement of a more or less complete climatography. In the appendix, models for climatic tables are presented, a table for easy calculation of the probable error, an auxiliary table for computing the equivalent temperature, and a table giving the numbers of the consecutive days of the year. The content and order of treatment in this book reflect, for the greater part, the experience the author acquired in his lectures in climatology and in supervising the theses of his pupils at the University of Vienna, Austria. A number of the examples are taken from the author's papers. It is with the greatest pleasure that the author expresses his gratitude to Professor Charles F. Brooks of Harvard University for his repeated encouragement to summarize in book form the methods of climatology. Thanks are also due to Dr. Brooks for his reading of the manuscript and many suggestions as well as for constant help in facilitating the routine work connected with this book. The author wishes to express his appreciation for financial assistance from gifts of the Associates of Physical Science and Anonymous Fund 15 of Harvard University. He is deeply indebted to Professor L. W. Pollak, Dublin, Ire- land, who himself has contributed so much to the problem of statistical methods in meteorology and climatology, for his reading the manuscript and his valuable comments and suggestions. His warmest thanks are extended to Professor W. M. Fuchs, New York, Dr. J. T. Morley, Dublin, Ireland, and Dr. C. Chap- man, Boston, who did their best to smooth his English. Grate- fully he acknowledges the assistance of Miss Helen Gilman for redrawing the figures. Mr. R. W. Burhoe, Acting Librarian of x METHODS IN CLIMATOLOGY Blue Hill Meteorological Observatory, provided the necessary literature, always very helpfully and promptly. Owing to the present war, it has been impossible to obtain permission from the respective scientists and publishers to repro- duce here a number of graphs. The author offers his sincere apologies for this omission and assures all concerned that, in normal times, he would have waited until he had received permission, before publishing. In this book, he has at least been careful to see that in the legend every figure taken from another work has been duly attributed to the scholar who originated it. V. C. Cambridge , Mass. October l t 1944 CONTENTS INTRODUCTION 1 PART I GENERAL METHODS I. CLIMATOLOGICAL ELEMENTS. COMPARABLENESS OF CLI- MATOLOGICAL SERIES 3 1. Climatological collaboration and organization. Sta- tions of different orders. The "rseau mondial" 5 2. The meteorological register 9 3. Some remarks about methods of climatological ob- servations 12 a. Discontinuities in climatological series 12 b. Observing air temperature. Errors caused by the influence of radiation and by layering of air close to the ground 13 c. Measuring precipitation 15 II. GENERAL STATISTICAL CHARACTERISTICS OF CLIMATOLOG- ICAL ELEMENTS 17 1. The system of characteristics 17 2. Primitive characteristics 17 3. The variate. The absolute extremes. The range of variation 17 4. Class intervals. Frequency distribution. Absolute and relative frequency 19 5. Histogram and frequency curve. The mode. The modal class 21 6. Elementary characteristics. Median, quartiles, dec- iles 24 III. SPECIAL STATISTICAL CHARACTERISTICS OF CLIMATOLOGICAL SERIES 30 1. The arithmetical mean 30 2. Deviations. Average variability. Standard devia- tion. Normal distribution 31 3. A practical application of the standard deviation in climatology 35 4. Variance. Coefficient of variation 38 5. Summary of statistical concepts 39 6. Relation between ju, cr, and f. Cornu's theorem. Unilaterally and bilaterally limited variates 39 xi xii METHODS IN CLIMATOLOGY 7. Higher characteristics (skewness) 44 8. Different kinds of variability used in climatology. . . 46 9. Absolute and relative variability and other measures of variations 51 10. Some applications of the method of random samples 53 a. Precipitation 53 b. Cloudiness and duration of sunshine 55 IV. SOME PROBLEMS OF CURVE FITTING AND SMOOTHING OF NUMERICAL SERIES 58 1. The straight line 58 2. Decay-curves 62 3. The equation y = B -e Ax 63 4. The quadratic equation 64 a. Level of the turning point of the curve 66 b. The standard distribution 67 5. Smoothing of numerical series 68 V. HARMONIC ANALYSIS 70 1. The analysis 70 2. Evaluation of the equation 76 3. Relative amplitudes 78 4. Times of the extremes 78 PART II REPRESENTATION OF CHARACTERISTIC FEATURES OF DIFFERENT ELEMENTS VI. TEMPERATURE 81 1. Hours of observation. Reduction to the "true mean" 81 2. Dates at which the average temperature crosses cer- tain thresholds 92 3. Duration of temperatures above and below certain thresholds 94 4. Other characteristics of temperature conditions. Spells of cold and hot days 95 5. Cumulated temperatures 96 6. Degree days 97 7. Relative temperatures 98 VII. ATMOSPHERIC PRESSURE AND PRESSURE OF WATER VAPOR 100 VIII. WIND 101 1. Resultant wind velocity calculated from records of direction and velocity 101 2. Resultant wind direction calculated from frequencies of the directions alone 103 3. The resultant run of the wind 104 4. The steadiness of the wind 106 CONTENTS xiii IX. SOME COMBINED ELEMENTS 107 1. Relative humidity 107 2. Equivalent temperature 108 3. Drying power Ill 4. Cooling power 112 X. CLOUDINESS 115 1. Clear and cloudy days 115 2. Fog 117 XI. PRECIPITATION 118 1. Probability of a day with precipitation 118 2. Probability that a day with precipitation is a day with snowfall 119 3. Amounts of precipitation within certain periods. ... 119 4. The annual course of the amounts of precipitation. . 120 a. The reduction to months of equal length .... 120 b. Amount of precipitation on an average day of the month 121 c. Ecart pluviom&rique relatif and the relative pluviometric coefficient 121 5. Absolute and relative variability. Quotient of varia- tion. Method of random samples applied to rec- ords of rainfall 122 6. Rain intensity 122 7. Wet and dry spells 123 8. Characterization of the hydrometeoric type of a period 1 24 9. Snow 124 10. Index of aridity 127 11. Rain-histogram 127 PART III METHODS OF SPATIAL COMPARISON XII. COMPARISON OF OBSERVATIONAL SERIES OF DIFFERENT PLACES. GEOGRAPHICAL DISTRIBUTION OF CLI- MATOLOGICAL ELEMENTS 129 1. Uniform tendency of the variations of average weather over large regions. Consequences drawn from this tendency for comparing series of ele- ments observed simultaneously at different places 129 2. Criteria of relative homogeneity 134 a. Helmert's criterion 134 b. Abbe's criterion 135 xiv METHODS IN CLIMATOLOGY 3. Reduction of climatological averages to a certain period 139 a. Method of differences 139 b. The reduction of precipitation series. Method of quotients 142 c. Physical explanation of the method of quo- tients 143 d. The reduction of other elements to a given period 144 e. Limits of the method of reducing climatological series to a normal period 144 f. Additional note. The length of a normal period 146 g. Interpolation of missing observations 148 h. Coherent and incoherent climatic regions and their statistical evidence 148 C. INTERMEDIATE CHAPTER. CORRELATION 150 1. Linear correlation 150 2. Example of calculating a correlation 153 3. Simplifications in computing correlations 155 4. Regression equation 157 5. Partial correlation \ 161 XIII. GRAPHIC COMPARISON OF CLIMATOLOGICAL ELEMENTS. ISOGRAM. MAP OF ISOLINES 166 XIV. METHODS OF ANOMALIES 170 1. Temperature distributions in mountainous regions. 170 2. Examination of lapse rate 170 3. Mapping temperature conditions of a mountainous country (Reduction to a given level) 171 4. Actual temperatures 172 5. Standard curves. Anomalies. Isanomals 173 6. Applications of the method of anomalies and isanomals 1 74 a. Vegetative period and altitude 1 74 b. Temperature and geographical latitude 177 c. Pleions, meions 178 d. Snow-cover and altitude 180 e. Duration of sunshine vs. elevation 180 f. Anomalies of precipitation 180 7. Precipitation profiles 183 8. Variability of precipitation and amounts of precipi- tation 185 XV. WIND 186 1. Wind roses 186 a. Wind roses for different elements 187 CONTENTS xv 2. Maps of streamlines 188 a. Streamlines based only upon frequencies of direction 192 XVI. AlR MASS CLIMATOLOGY 193 XVII. NUMERICAL CHARACTERIZATION OF DIFFERENT CLIMATIC FEATURES 195 1. Continentality 195 2. Limits between timber-forest, steppe, and desert. Effectiveness of precipitation 196 PART IV THE CLIMATOGRAPHY XVIII. ARRANGEMENT OF A CLIMATOGRAPHY 199 1. The introduction 199 2. Static climatology 200 3. Monographs 201 4. Records of self-registering instruments 202 5. Change of temperature and precipitation with height 202 6. Graphical representation 202 7. Dynamical climatology 206 8. The bioclimatological part 212 9. Description of climatic phenomena 212 XIX. CONCLUSION 215 APPENDIX I. Forms of individual climatic tables 216 APPENDIX II. Vl/n 218 APPENDIX III. Auxiliary table, calculating the equiva- lent temperature from air temperature and relative humidity 219 APPENDIX IV. The days of an ordinary year, numbered consecutively, beginning January first. 220 INDEX 221 TABLES 1. Daily temperature minima at Mt. Washington, N. H., for four months of January 18 2. Frequency table and class intervals 19 3. Frequency distributions characterizing different climates 20 4. Example of an array and grouped data 24 5. Frequency distribution for a larger sample 25 6. Dates of the last killing frost. Deviations 32 7. Classification of temperature minima based on the probable error 37 8. Statistics of rainfall in March at Helwan, Egypt 42 9. Examples of average variability and inter-sequential varia- bility 48 10. Different statistics regarding interdiurnal variability of tem- perature 49 11. Method of random samples applied to rainfall 55 12. Examples illustrating the relation between duration of sunshine and cloudiness 56 13. Annual amplitude of air pressure and altitude 58 14. Example of observations represented by an exponential curve. 63 15. Example of observations represented by a quadratic equation. 65 16. Clipping from a table of normal distribution of temperature in a mountainous country 67 17. Instruction for calculating the harmonic constants for 12 equi- distant values 73 18. The same for 24 equidistant values 73 19. Example of the harmonic analysis of an annual course 76 20. Evaluation of the resulting equation 77 21. Example of a daily course of temperature 82 22. Hours of observations at climatological stations in the United States 84 23. Examples of the annual course of the correction to give the "true mean" for different hours of observation 85 24. The correction term of the arithmetical mean of the average extreme temperatures to give the "true mean" 89 25. The annual course of temperature at Bismarck, N. D 93 26. Average cumulated daily minimum temperatures for Paris. ... 96 27. Examples of degree days 97 28. Examples of the calculation of the wind elements 101 29. Equivalents of the numbers of the Beaufort scale 105 XVll xviii METHODS IN CLIMATOLOGY 30A. Average monthly equivalent temperatures at Boston, Mass., computed by the approximating formula 110 30B. Comparison between the approximating and the correct formula 1 10 31. Frequency of clear and cloudy days and cloudiness 116 32. Probability that a day with precipitation is a day with snowfall 119 33. Representation of the annual course of precipitation 120 34. Form for representing snow conditions 125 35. The quasi-constancy of differences of temperature at two differ- ent places 131 36A. The quasi-constancy of the ratios between rainfall at two differ- ent places 132 36B. Fluctuations of rainfall amounts as percentage of the average values as illustrative of the quasi-constancy of ratios (BjW) (after L. W. Pollak) 133 37. Example of the application of Abbe's criterion to differences of temperature 137 38. The same for ratios of rainfall 137 39. Temperature conditions for different periods at St. Paul, Minn. 139 40. Rainfall on the Maiden Island during different periods 139 41. Example of calculating a correlation 154 42. Calculating a correlation by means of the original numbers diminished for a constant value 156 43. Another example of calculating a correlation 159 44. Standard distribution of the duration of the vegetative period in Switzerland 175 45. Examples of anomalies 175 46. Standard distribution of temperature with geographical latitude 177 47. Example of wind roses 188 48. Degree of continentality derived from frequencies of continental and maritime air masses 193 ILLUSTRATIONS FIGURE 1 . Structure of a climatological network 6 2. The international climatological register 10 3. The register of the cooperative stations (U.S.A.) 11 4. Frequency histogram and curve 22 5. Normal distribution and standard deviation 34 6. Frequency distribution of daily minimum temperatures. ... 38 7. Frequency curve of a unilaterally limited variate 43 8. Frequency distribution of a bilaterally limited variate 43 9A. A right-skewed frequency distribution 45 9s. A left-skewed frequency distribution 45 10. Relative variability and rainfall 52 11. Cloudiness and duration of sunshine 55 12. Annual range of air pressure and altitude 59 13. Observations represented by the equation y = Ax~ B 62 14. Observations represented by the equation x= A + By + Cy 2 64 15. Superposition of two waves 71 16. Determining the true phase angle 75 17. Evaluation of a Fourier-series of two terms 77 18. Daily course of temperature 82 19. Isolines of the correction of the averaged daily extremes of temperature to give the " true mean M for the United States in January 90 20. The same for July 90 21. Interpolating the dates at which the annual temperature curve crosses certain thresholds 92 22. Relative temperatures 98 23. Evaluating the true azimuth of the resultant wind 103 24. Frequency of wind directions and their average velocities. . . 104 25. Climogram of Boston, Mass 107 26. Variations of January temperatures at Moscow 130 27. Extreme range of temperature and length of the period of observation 147 28. Dot-diagram and regression lines 161 29. Isopleths 166 30. Isanomals of the duration of the vegetative period in Switzer- land 176 31. Isanomals of January temperatures 179 32. Isanomals of sunshine duration for the eastern Alps in winter 181 33. Profile of precipitation across the Bernese Alps 184 34. Profile of snowfall across New England 185 XIX xx METHODS IN CLIMATOLOGY FIGURE 35. "Wind-star" 187 36. Streamlines over the Balkans 189 3?A. Streamlines of New England in January 190 37e. Streamlines of New England in July 191 38. Isolines of the dates when the annual temperature curve exceeds 70F in the Mediterranean region 204 39. Isolines of the dates when the annual temperature curve drops below 70F in the Mediterranean region 204 40. Isolines of the duration of 70F in the Mediterranean region 205 41. Isolines of the duration of the dry period in the Mediter- ranean region 205 42-45. Distribution of the basic elements characterizing normal, cold, and warm seasons in the United States 207-210 46. Eroded slopes in the mountains of Albania facing p. 212 METHODS IN CLIMATOLOGY INTRODUCTION CLIMATOGRAPHY describes the climate, climatology explains it. Climatography prepares the raw material supplied by ob- servation; it is the foundation of climatology. This seems a vicious circle, because it is hardly possible to make a climatog- raphy without a knowledge of climatology. This is the present state in which the two branches of science supplement each other. In its early beginning, climatography was an occasional description in words and pictures by travelers, chroniclers, writers, poets, and painters. The invention of the thermometer and its regular use for ob- serving air temperature was the first step toward a quantitative climatography. Galileo Galilei invented the thermometer in 1597. It became an accurate physical instrument in 1780 when de Luc introduced mercury as a thermometrical substance. Rain gauges are supposed to have been used in India in the fourth century B.C. The oldest regular quantitative measurements of rain were made in Palestine in the first century A.D. With these observations at hand, it has been possible to compare the hydrometeoric climate of ancient Palestine with the present one in an exact way. The invention and development of other instruments has furthered regular quantitative climatological observations. Elements which cannot be observed by instruments are estimated by means of arbitrary scales. A single observation, picked from a series, has usually no clima- tological meaning. Only the chronological arrangement and, gen- erally, the combination of observations, are sufficient basis for computing average variations and average states. A single ob- servation can be compared to a single exposure of a movie film, which has no real meaning until it is combined in its proper sequence with the rest. The nature of our subject determines the necessary analytical and synthetical methods. First of all, the meaning of climate may be explained : Climate is the average state of the atmosphere above a certain place or region of the earth's surface, related to a certain epoch and consider- is subject. 1 Observations are made at isolated points. Only by comparing these data can the climate of the whole region be interpolated. It is therefore the principal and fundamental aim of climatological methods to make the climatological series comparable. The more fully this goal is approached the more reliable are the indications of a climatography. Long-range forecasts or investigations in hidden periodicities depend even more on reliability and exactness, and therefore upon the comparableness of climatic series. Only with comparable climatic series can true averages, true variations, true probabilities be determined and used for making further estimates and inferences. 1 V. Conrad, " Die klimatischen Elemente und ihre Abhangigkeit von terrestrischen EinflQssen," Handbuch der Klimatologie, ed. by W. Koppen and R. Geiger, vol. IB (Berlin, 1936). PART I GENERAL METHODS CHAPTER I CLIMATOLOGICAL ELEMENTS. COMPARABLENESS OF CLIMATOLOGICAL SERIES CLIMATOLOGICAL ELEMENTS result from the analysis of the state of the atmosphere. The combination of all elements occurring at a given moment makes the weather; the average weather means the average state of the atmosphere, that is, the climate. It is not possible to enumerate all the elements, because their number can be increased arbitrarily, but the following may be listed : 1) Radiation of sun and sky 2) Temperature of the air and of the surface of the earth 3) Wind direction and velocity 4) Humidity and evaporation 5) Cloudiness and sunshine 6) Precipitation 7) Snow cover 8) Air pressure, because of its intimate relation to the instantaneous and average state of weather and atmosphere. The difficulty is that these items do not represent single ele- ments, but groups of elements. The radiation of sun and sky, as an example, is divided into two main groups: direct radiation from the sun, and radiation from the sky. Each group can be subdivided into any number of elements. There is the total radiation, including the energy of the entire spectrum, which can be divided into the elements characterized by the energy of parts of the spectrum, for example the infrared, the visible, the ultra- violet portions. Other radiation elements deal with the trans- mission coefficients of certain wave-lengths, with the turbidity factor, the blueness of the sky, photometrical and photochemical 4 METHODS IN CLIMATOLOGY brightness, atmospheric back radiation, range of the ultraviolet spectrum, etc. 1 The group of radiation represents only one example. The same conditions exist with all the other groups. This nearly infinite multifariousness is no misfortune. On the contrary, a clever climatographer, while utilizing the usual elements, should define new elements which give the best, the shortest, and the most accu- rate contribution to the description of the climate in consideration. For the present purpose, it is perhaps of greater importance to classify the elements from another point of view : 1) Primitive elements, directly observed or estimated 2) Combined elements 3) Derived elements. Examples of group 1, primitive elements, are temperature, precipi- tation, wind velocity, wind direction, cloudiness, etc. Examples of group 2, combined elements, are turbidity factor, continentality, equivalent temperature, potential waterpower, etc. There are also elements which may be taken both as primitive and combined elements; examples are relative humidity and cooling power. On the one hand, each can bfc read directly from its ap- propriate instrument; on the other hand, each can be calculated by means of a combination of two or three other elements. The third group is obviously the most extensive and offers much scope to the imagination of the climatographer and climatologist. Only a few examples may illustrate the importance of this group. There are the variabilities of the different elements, the length of certain spells, as of high and low temperatures, of dry or wet weather, etc. ; the duration of characteristic average states of the atmosphere for example, the duration in days of the period during which the average temperature is below the freezing point (frost period) or the duration of the vegetation period, that is the time during which the average temperature is ^43F; the lapse rate of temperature; the manifold use of frequencies and probabilities, anomalies, etc. From the classification of the elements as primitive, combined, and derived, their infinite multiplicity is obvious; 2 hence the need 1 For the multiplicity of radiation elements, see the monograph of F. W. P. G6tz, Das Strahlungsklima von Arosa (Berlin, 1926). From the same point of view, W. M6rikofer, " Meteorologische Strahiungs Messmethoden," in Emil Abderhalden, Hand- buck der biologischen Arbeitsmethoden (Berlin, 1920 ) Abt. II, pt. 3, is of interest, 1 In the foregoing series of climatic elements, the one or the other is enumerated which may be unknown to the reader. Some of them are explained in the following CLIMATOLOGICAL ELEMENTS $ of some methodical hints on handling these numerous expedients and selecting the best in order to get the most concise and exact description of the climate. Observations do not always reproduce natural conditions in the right way. There are many sources of error. An appropriate critique of the data should eliminate the systematic as well as random errors. These corrections concern a series of observations made at one place for a period of time, and a comparison of observations made simultaneously at different places. The preparation of original observations for purposes of com- parison, whether as single values or as a series, is one of the main tasks of climatological methods. I. 1. CLIMATOLOGICAL COLLABORATION AND ORGANIZATION. STATIONS OF DIFFERENT ORDERS. THE "RESEAU MONDIAL." Climatology is a world-wide science. That is clear from the standpoint of comparability. The observations of a station A have to be compared with those of station B south of A ; the ob- servations of C west of A should be compared either with A or with B, and so forth, for all directions of the compass. (See Fig. 1.) Thus we may understand why Sir Napier Shaw writes in his Manual of Meteorology (I, 160) :"No country, not even the largest, is self-sufficient in the material for study, because that material must be co-extensive with the whole world." The first series of observations were made by private men and institutions interested in weather and nature. In 1781, the meteor- ological organization the " Mannheimer Akademie" was founded; and this was the first step toward a world-wide climatography. In their publications, Ephemerides Societatis Meteor ologicae Palatinae, the results of observations for two American places were also published. Nowadays, in civilized countries, the State has organized central institutes which administer and extend the network of climato- logical (meteorological) stations. The expression " network " is taken from the "network of stations " upon which the geodetic survey is based. Figure 1 indicates more fully the meaning of the chapters in so far as they are of methodological interest. Other explanations can be found in V. Conrad, Fundamentals of Physical Climatology (Harvard University Press, 1942) and in V. Conrad, "Die klimatologischen Elemente und ihre Abhangigkeit von terrestrischen EinfKissen," Kdppen-Geiger, Handbuch der Klimatologic, vol. IB (Berlin, 1936). 6 METHODS IN CLIMATOLOGY term. The observations of each station have to be comparable with those of the others. The aim of the study of the reciprocal relations between the climatic elements at different places is, on the one hand, to determine the circulation of the atmosphere (physics of the actual state of the atmosphere), and, on the other, to get full information about average and extreme values of clima- tological elements at least on the earth's surface (physics of the average state of the atmosphere). FIG. 1. Structure of a climatological network For these large-scale purposes, a network, internationally ad- ministered, has to cover the entire globe. Regional meteorological efforts found their international organization at a relatively early stage. An international conference of maritime meteorology was held at Brussels in 1853, ninety years ago, and the first international meteorological congress at Vienna (Austria), in 1873, was attended by official delegates from a large number of countries. On this occasion an " International Meteorological Committee " was elected in order to assure international meteorological and climatological collaboration without interruption during the interval between the congresses. This international organization has the following duties : 1) To make regulations and offer advice concerning com- parable observations. CLIMATOLOGICAL ELEMENTS 7 2) To establish definitions of phenomena like rime, snow- cover, day with thunderstorm, day with precipita- tion, etc. 3) To make regulations regarding the organization of a network for a country. Such a network should consist of stations of different orders : (a) A Central Office, or Central Institute, is the chief office en- trusted by the Government with the management, collection and publication of the meteorological observations of the country [e.g. the U. S. Weather Bureau]. (6) A Central Station [Section Center] is a subordinate center for the management and collection of observations from a certain province [state; e.g. the U. S. Weather Bureau, Boston, Mass.]. (c) A Station of the First Order is an observatory in which, with- out the collection of observations from other stations, meteor- ological observations are conducted on a great scale, i.e. either by hourly readings or by the use of self-recording instruments. (d) Stations of the Second Order are the stations where com- plete and regular observations on the usual meteorological ele- ments, viz., pressure, temperature and humidity of the air, wind, cloud, rain and hydrometeors, etc., are conducted. (e) Stations of the Third Order, finally, are the observing sta- tions, where only a greater or less portion of these elements are observed. 3 A knowledge of the international organization, of its regulations and its definitions, is of great importance to anyone who has to use the meteorological records of other countries. The most es- sential resolutions and recommendations of the International Meteorological Committee during the years from 1872 to 1905 are summarized in the International Meteorological Codex. 4 In spite of the age of this report (published in English by the British Meteorological Office in 1909), the recommendations are valid and valuable today with the exception of the sections on aerological observations and on weather maps and forecasting. A Spanish version was published by the Observatorio Central de Manila in 1913, and includes the resolutions adopted between 1872 and 1910. 5 Another merit of the International Meteorological Organiza- tion is the publication of the reseau mondial, a global network of meteorological stations. This title represents the climatological and meteorological ideal: a network (as stated above), which 1 Sir Napier Shaw, Manual of Meteorology, I (1926), 163. 4 Prepared by G. Hellmann and H. H. Hildebrandsson. *Sir Napier Shaw, Manual of Meteorology, I, 166. For further information, see: Official Reports of the International Conferences. 8 METHODS IN CLIMATOLOGY covers the entire surface of the earth, administered in a uniform manner and devoted to all of mankind. The publication in ques- tion is only the beginning of an ideal as yet unaccomplished. It was a great scientific advance when the British Meteorological Office initiated it in 1911. Monthly averages of pressure, tem- perature, and precipitation at about five hundred places are pub- lished for each year. The stations are grouped in 10-zones of latitude. It is noteworthy that meteorological data can be sup- plied for all the zones from 80N to about 61S. South of this latitude no permanent meteorological station is being operated. The data and elements given for each station in the rSseau mondial for the months and the year are : 1) Latitude 2) Longitude 3) Height (meters) 4) Pressure (millibars): (a) Mean at the level of station (b) Mean at mean sea level (M.S.L.) (c) Differences from normal 5) Temperature in the absolute scale, starting from 273C below the freezing point of water: (a) Mean maximum (b) Absolute maximum with date of occurrence (c) Mean minimum (d) Absolute minimum with date of occurrence (e) Arithmetical mean of (a) and (c) (f) True average temperature (g) Difference from normal 6) Precipitation: (a) Monthly amount in millimeters (b) Difference from normal Another document of the international development of clima- tology and meteorology is the World Weather Records, assembled and arranged by H. Helm Clayton, in consequence of a resolution of the International Meteorological Conference adopted at the meeting of Utrecht in 1923. This publication 6 gives for each year of record the monthly and annual values of air pressure, tempera- ture, and precipitation, and averages of all, for about 390 places. e Smithsonian Miscellaneous Collections, vol. 79 and vol. 90, Washington, D. C. 1927 and 1934. CLIMATOLOGICAL ELEMENTS 9 The great advantage is that the publication offers long and, as far as possible, homogeneous series, which overlap one another for the period of observation. Thus the work is indispensable if variations of the three elements have to be compared. I. 2. THE METEOROLOGICAL REGISTER The meteorological register contains the records of the original, regular observations made at the climatological station. All cli- matological studies are based ultimately upon the registers of the stations. The utmost international uniformity of the registers would be advantageous for all climatological investigations which cover a region larger than one national network. This was the reason why the International Meteorological Organization (Utrecht, Hol- land, in 1874) suggested a special form for the register of stations of the second order. Figure 2 shows this international form with a few supplements. As far as the units of pressure and altitude, the temperature scale, etc., are concerned, uniformity could not be achieved. This form, which assumes that the elements mentioned are observed three times a day, is used by most of the central institutes of Europe, of the U.S.S.R., etc. In many networks, the register is more or less modified, according to the various conditions and requirements. It is to be hoped that these modifications will be restricted more and more, and will give way to complete uniformity. As far as the organization of the United States Weather Bureau is concerned, J. B. Kincer may be quoted: 7 The Weather Bureau has maintained some 200 observing stations known as stations of the first order, manned by professional meteor- ologists, at which complete meteorological observations are made, in- cluding automatic, continuous instrumental records of many weather elements. Since these stations are not spaced closely enough to provide data adequate for climatic purposes, a large number of cooperative stations . . . are maintained to obtain the necessary basic data. [The number of these cooperative stations is about 5,000.] On the other hand, the U. S. Weather Bureau has branches in the States which administer the respective climatic section and are equipped with many instruments, especially automatic-recording instruments. 7 "Climate and Weather Data for the United States/' in Climate and Man (Year- book of Agriculture, United States Department of Agriculture, Washington, D. C. f 1941), p. 689. 10 METHODS IN CLIMATOLOGY II H 4? 43 43 c a 4) S o S l a 43 43 43 43 43 43 43 Remarks about hours and duration of precipitation, thunderstorms, storms, optical phenomena, etc. d^ ^* 43 &g 4- ^ Q *J S !& a '4* 1 cl crJ <n & O i* N 'S'Sl ^ rt S 43 ^ . ** 43 - 4) t db^ ~ 43 43 43 Q -*.. bA 4> III O I 4) d X) rt ri u u, Jj s-s i ! 4^ ill 4) 4) Ifl-S 43 4343 4) 4) KB u, 0) bo 4> O S CJ "i ^o rt G <D ctJ ^ pq CN 6 CLIMATOLOGICAL ELEMENTS 11 Because the network of the cooperative stations of the U. S. Weather Bureau is one of the greatest in the world, it may be con- venient to give here the register of these stations as an example of a somewhat modified international form (Fig. 3). U. S. DEPARTMENT OF COMMERCE, WE ATHER BUREAU COOPERATIVE OBSERVERS' METEOROLOGICAL RECORO snnont <fr*tfri//f . . , C(. t ATirunr44 f 23 A/ LM9/ruOf74'27W fLCUirtO*. ttONTM WA*/- , 1934. HfltlDtAN Of TIME. t-Jte 27. 'MT 21 MONTHLY SUMMARY MICH WINOt, JffM. ,. GlAZt. tmttiiM IUINI.. KltUNOfKOftT,. TMONOCMSTOMMS, HtfL MMC " kdt f M i /J//T..,, OCPTH Of rK02CN OMOUNO AT CNO Of MO _W1 ^3*5*7 29 4i 1P- SI JLiL J^ (MOt TO M riCtCO M OtMHVl*) TEMPERATURE MfAN MAIMUM MCAN MINIMUM ^ MEAN MONTMtt. DCWkHTW MAXIMUM MINIM 16 TOTAt.- 9 PRECIPITATION INS OC*AaTU*f,_ QNCATCST AMOUNT f* 47 35 1.5 TOTAL SNOfTfALt. MCATCST IN ANY 00SCftV*TlONAt OAT. 8 M- tl+r **/* A fO* /M = *. NUMOEROPOAYS MAX TCM^ O* **ft^> MAX TCM^ Jl 0* "*. MIN TtM^ S ON ItLO* MIN TCMP o* o* ICLOI WITH 01 INCH 0* MOC PKCCl^. l INCH on MOKE, . IOO tNCH Oft MOftC, CLt A. MUTLT CLOUOT,___ttOUOt. FIG. 3. Form of the register of the cooperative stations of the U. S. Weather Bureau The greatest difference between the two forms lies in the fact that the international form is arranged for three observations a day, while the U. S. cooperative observer makes one observation a day. Two columns are reserved for the extremes of temperature. A third column of this section contains numbers which are not of interest here. A great advantage, not provided in the interna- tional form, is the columns which indicate the time of the be- ginning and end of precipitation. Fundamentally important differ- ences are noted in the case of cloudiness and wind : here the prin- 12 METHODS IN CLIMATOLOGY ciple of making observations of an instantaneous state is not maintained, estimates of an average state during the day preceding the observation being given instead. The remarking of differences does not imply, of course, a criti- cism; a scientific organizer is free to arrange the methods of ob- servation so that he gets the greatest number of features character- istic of weather and climate. The student who builds up his climatography on the registers should be conscious of these facts, even if he is not compelled to go back to these fundamental sources. It might happen, for instance, that the region whose climate has to be described is crossed by the borderline between two clima- tological networks. In the first of these, the average direction of the wind during the day is estimated; in the second, momentary wind directions are observed three times a day. It would be a source of errors if the different methods of observation escaped notice. Usually, a space in the form is reserved for some simple sta- tistical values which can be easily derived from the original observations. I. 3. SOME REMARKS ^BOUT METHODS OF CLIMATOLOGICAL OBSERVATIONS The following discussion is not intended as an instruction on the arrangement of climatological observations. It serves only to emphasize difficulties of comparison which may arise with different elements. I. 3. a. Discontinuities in Climatological Series When the variations with time of an element at a place have to be investigated, its values at different times are compared. Changing the method of observation can cause a break, a discon- tinuity of the series. If a thermometer of the ordinary type is read over a period of ten years and is then replaced by a sling- thermometer, the temperatures during daytime become lower and night temperatures become slightly higher. Thus a discontinuity is caused, so that the second series cannot be compared with the first. The paradox is that while the sling-thermometer means a real improvement, this improvement destroys the continuity of the series. If after the introduction of the new procedure parallel observations are made in the old and in the new manner over a sufficiently long period, the values of the old series can be reduced CLIMATOLOGICAL ELEMENTS 13 to those of the new one. Suitable methods for such reduction are given in the following chapters. An example of a nearly ideal "secular" station is represented by the Blue Hill Meteorological Observatory, south of Boston, Mass., founded by Abbott L. Rotch in 1885. Here observations have been made with suitable devices according to well-considered methods from the start, fifty-eight years ago. While numerous more modern methods were being introduced, the old methods of observing the most fundamental elements were continued simul- taneously with the new ones. No change has taken place in the surroundings. I. 3. b. Observing Air Temperature. Errors Caused by the Influence of Radiation and by Layering of Air Close to the Ground Observing the air temperature in order to make comparisons involves a considerable number of difficulties. Only one example need be mentioned here, which may contribute to a better under- standing of the later sections of this book. A well-made ther- mometer shows the "true temperature of the air M if its bulb is in full heat-conductivity balance with the passing air. This ob- viously simple demand can hardly be fulfilled in free air because of the radiation from sun and sky, from the ground and from sur- rounding objects. The invention of the "aspiration-thermometer " eliminated the errors caused by radiation, for meteorological pur- poses. Because of some difficulties which need not be discussed here, climatological air temperature is defined by the indications of a well-constructed aspirated thermometer at the level of about 5 feet above the ground. Thermometers sheltered in a suitably placed "Stevenson screen " show temperatures which, generally, equal the indications of an aspiration-thermometer, so that the difference can be practically neglected. Thus, but for the existence of another difficulty, the problem of observing the climatological temperature would have been solved. In the Pamirs (Central Asia, about lat. 38N), in summer, a difference of about 60 F 8 was observed between the temperature of the layer adjacent to the ground, and that of the 5| ft level above it. 9 Once in winter, in a depression in the eastern Alps, a The author gladly introduces a suggestion of Charles F. Brooks to discriminate in printing between F, C = an actual temperature and F, C = Fahrenheit, Centi- grade degrees representing a temperature difference. 9 V. Conrad, "Die klimatologischen Elemente und ihre Abhangigkeit von terres- trischen Einfltissen," K6ppen-Geiger, Handbuch der Klimatologic, IB, p. 72. 14 METHODS IN CLIMATOLOGY temperature of 20F was observed on the ground and, simul- taneously a temperature of +30F at the 180 ft level above the ground. 10 The physical explanations of these conditions, interesting as they are, need not be set forth. Here, only the fact counts that thermal stratifications of enormous intensity exist within thin layers above the ground. Such abnormalities, though not of this in- tensity, are frequent everywhere. We know this from an investi- gation in an exceedingly equable climate, near London, England. 11 From this study we also learn that the average vertical tem- perature gradient in Kew (England) is so great that the tempera- ture decreases 1 F per 3J inches, within a layer from 1 ft to 4 ft above the ground, in June. This is a gradient 625 times greater than the dry-adiabatic gradient. The stratification of the air according to its specific gravity is, however, a very common climatic phenomenon in the concavities of the surface, when the outgoing radiation exceeds the incoming. The stratification is observed for a concave surface of any dimen- sion. It appears in the broad and narrow valleys of the high mountains of the earth, and is experienced in every furrow of the field. Thus the layering of the f air is of interest, not only for general considerations in observing air temperatures, but also as one of the most important features from which "Microclimatol- ogy," and especially "Orographic Microclimatology," makes its start. 12 And now a problem arises concerning the temperature-observa- tion : namely, to which layer should the observation of temperature be related, if the temperature varies in the lowest layers for every 3^ inches in the vertical direction as much as in the horizontal for about one degree of latitude. The observations with the aspiration- thermometer, especially, become very inexact and ill-defined. Temperatures vary several degrees, according to the direction from which the air is sucked up. There is only one expedient. The place of observation at a climatological station should be a spot with good natural ventila- tion, open to all wind directions, so that thermal stratification is rare. Furthermore, the thermometer-bulb should not be too close to the ground. The International Meteorological Organization 10 Ibid., p. 183. 11 A. C. Best, Transfer of Heat and Momentum in the Lowest Layer of the Atmosphere (Geophysical Memoirs of the Meteorological Office, London, No. 65, 1925). 12 R. Geiger, Das Klima der bodennahen Luftschicht (Braunschweig, 1927). See especially the second part: " Orographische Mikroklimatologie," and Figure 30, p. 98. CLIMATOLOGICAL ELEMENTS 15 advises fixing the thermometers at a height between 5 and 6| feet. The thermometers in the screen of cooperative observers of the U. S. Weather Bureau are at about the height mentioned, and thus exclude the bottom layers with intense, vertical temperature gradients positive or negative. We learn from this example that the exposure of the thermo- meter shelter must be chosen so as to avoid, if possible, local influ- ences of radiation and thermal stratification. Generally, only the former influence is mentioned, but the latter is also a great dis- advantage. Both factors seriously impair the comparableness of the data. The thermometer shelter should be located so that the recorded temperatures are representative of the largest possible surrounding area, a condition which also makes for the most com- parable results. It is obvious how badly the continuity of a series of temperatures may be disturbed by shifting of the shelter. I. 3. c. Measuring Precipitation Measuring precipitation offers a difficult problem. None of the various methods gives entirely satisfactory results. The amount of water or snow collected in the gauge depends, to a certain degree, upon the wind velocity and the air resistance of the particles of precipitation. The stronger the wind and the greater the air resistance to the particles, the smaller is the amount caught in the gauge compared with that collected under otherwise similar condi- tions during absolute calm. Shielded gauges, invented by F. E. Nipher, give incomparably better results. 13 In general, the normal climatological stations are not yet equipped with shielded gauges, so that the wind and the different types of precipitating particles still influence the measurements: hence, a degree of comparableness can be achieved only by using identical gauges installed in a uniform way. This is true for the United States, half a huge continent, \ here the exposure of rain-gauges follows sound uniform principles. If some reader is interested, however, in a study of precipitation in Europe, he is confronted with much more complex conditions. 18 For further information, see: C. F. Brooks, "Windshield for Precipitation Gauges" (Transactions of the American Geophysical Union, 1938, p. 539); C. F. Brooks, "Further Experience with Shielded Precipitation Gauges on Blue Hill and Mount Washington" (ibid., 1940, pt. II, p. 482); H. S. Riesbol, "Report on an Exploratory Study of Rain Gauge Shields and Enclosures at Coshocton, Ohio" (ibid., p. 474); F. N. Denison, "A Report on the Difference Between the Precipitation Records as Taken on the Standard Canadian and U. S. Rain-Gauges" (Bulletin of the American Meteorological Society, February 1941, p. 65). 16 METHODS IN CLIMATOLOGY From the following table, the areas of the receivers of types of rain-gauges used in some countries in Europe may be seen, and for comparison, the area of that used by the U. S. Weather Bureau : Austria 78 sq. inches England 31 sq. inches type I England 20 sq. inches type II France 62 sq. inches Germany 31 sq. inches U. S. Weather Bureau 50 sq. inches Perhaps the size of the receiver has no great influence upon the re- sults. In any case, these numbers show that there is no uniformity. As was said above, measurements of precipitation depend on wind velocity. Because this varies enormously with the height above the ground, it is clearly a scientific requirement that the rim of the receiver should be at the same height in all countries. In reality, the heights are: Austria 59 inches England 12 inches France 59 inches Germany 39 inches U. S. Weather Bureau 34 inches This is a violation of the scientific requisite of uniformity. The climatologist who has the task of giving the distribution of pre- cipitation over a region should be aware of these inequalities and discontinuities. CHAPTER II GENERAL STATISTICAL CHARACTERISTICS OF CLIMATOLOGICAL ELEMENTS THE INTERNATIONAL MODEL of the meteorological (climato logical) register, covering the period of a month, contains about 750 observations; 9,000 observations for the year. The registers of a small climatological network with one hundred stations yield 900,000 observations in one year. The 5,500 co- operative stations of the U. S. Weather Bureau produce registers with about 12 million entries per year. The human brain is not capable of comprehending such a huge mass of numbers. Thus, a statistical treatment of observational data is necessary. II. 1. THE SYSTEM OF CHARACTERISTICS The following pages present a selection of such methods of mathematical statistics as are suitable for abstracting the common characteristics of climatological elements. L. W. Pollak has made a simple survey of these characteristics, and his system is based on the amount of mathematical operations necessary to compute them. They are divided into groups: l 1) Primitive characteristics 2) Elementary characteristics 3) Higher characteristics. II. 2. PRIMITIVE CHARACTERISTICS Primitive characteristics can be derived from the original ob- servations by direct copying, by grouping under special headings, by counting, and by similar primitive methods. Examples will be given to pave the way to better understanding and to illustrate the close connection with climatological topics. II. 3. THE VARIATE. THE ABSOLUTE EXTREMES. THE RANGE OF VARIATION An example is the January minimum temperature at Mount Washington, N. H., for the four years 1938 to 1941. 1 L. W. Pollak, Charakteristiken der Luftdruck-Frequenz-Kurven und verallgemein- erte Isobaren, Prager Geophysikalische Studien, vol. I (Prague, 1927). 17 18 METHODS IN CLIMATOLOGY First, a list of the data, as in Table 1, is required. In this list, the extremes of the minima are easily checked for each year. It is practicable to italicize the maxima, and to mark the minima by asterisks. Then the extremes of the entire sample are seen at first TABLE 1. DAILY MINIMA OF TEMPERATURE (F) AT MOUNT WASHINGTON, N. H., FOR FOUR MONTHS OF JANUARY (Mount Washington Observatory News Bulletin) January 1938 1939 1940 1941 1 2 Q -14 6 2 5 -10 -14 9 3 1 -21 -10 9 4 8 2 -11 6 5 16 - 3 - 2 6 1 19 -14 - 2 7 15 20 - 6 1 8 - 3 - 2 9 -13 6 10 -10 16 - 3 11 -10 2 7 12 1 -13 5 - 2 13 4 -12 11 -30* 14 - 4 - 4 8 -30* 15 4 - 8 4 -24 16 - 5 6 -30* - 1 17 - 5 -15 -30* 19 18 -15* -15 -28 21 19 - 2 -16 -12 20 2 - 4 -15 -12 21 4 - 4 -11 - 6 22 8 2 -19 12 23 9 -36* - 8 - 6 24 11 -10 4. 25 19 -22 - 9 3 26 - 5 -29 -14 - 5 27 -13 -13 -12 6 28 -11 -12 -13 5 29 -13 8 - 8 -12 30 - 4 4 - 8 - 1 31 - 6 - 1 - 5 - 4 m - 1.1 - 4.7 - 8.7 - 1.5 -4.0 glance. These are the absolute extremes of the sample of the variate, that is the variable quantity. In Table 1, the variate is repre- sented by the daily minimum temperatures for the months of GENERAL STATISTICAL CHARACTERISTICS 19 January. We get : Absolute maximum = 21F Absolute minimum = 36F Difference = 57 F This difference is called the range of variation, and defines the entire range of the sample. The range is divided into a number of equal class intervals. II. 4. CLASS INTERVALS. FREQUENCY DISTRIBUTION. ABSOLUTE AND RELATIVE FREQUENCY The choice of the size of the class intervals is dependent upon the number of observations included in the sample and on the degree of exactness required. For the conditions of the model in Table 1, 4F-intervals are more or less suitable, because the sample consists of only 124 elements. TABLE 2. FREQUENCY TABLE. GROUPING THE DATA INTO CLASS INTERVALS OF 4 F Class Frequency Number (*) Limits ( [F) Center Absolute Percentage (%) (1) (2) (3) (4) (5) 1 -36 to -33 -34.5 1 0.8 2 .... -32 to -29 -30.5 5 4.0 3 .... -28 to -25 -26.5 1 0.8 4 .... -24 to -21 -22.5 3 2.4 5 - 20 to -17 -18.5 1 0.8 6 .... - 16 to -13 -14.5 15 12.1 7 -12 to - 9 -10.5 16 12.9 8 - 8 to - 5 - 6.5 13 10.5 9 .... 4 to - 1 - 2.5 19 15.3 10 to 3 1.5 19 15.4 11 4 to 7 5.5 13 10.5 12 8 to 11 9.5 9 7.3 13 12 to 15 13.5 2 1.6 14 16 to 19 17.5 5 4.0 15 20 to 23 21.5 2 1.6 * For some statistical purposes it is more desirable to begin with no. (zero). See: E. Czuber, Die statistischen Forschungsmethoden (Wien, 1921), p. 91; and L. W. Pollak, Prager Geophysikalische Studien, vol. I, pp. 33 and 34. These class intervals are seen in the second column of Table 2. In many cases it is more advantageous to give the central height or midpoint of the class interval, as in column 3. The first column contains the serial numbers of the class intervals. 20 METHODS IN CLIMATOLOGY The number of observations falling within the limits of the consecutive intervals is counted. The resulting series represents the frequency distribution of the sample in question. Column 4 in Table 2 gives the frequency distribution of minimum temperatures, etc. These numbers are called absolute frequencies. For funda- mental climatological purposes of comparison, relative frequencies are more expedient ; these are absolute frequencies expressed in per cent of the total number (n) of observations. (In the example of Table 2, n = 124.) Examples of the great usefulness of frequency distributions, which often mean an indispensable supplement to the usual averages, can easily be given. The figures in Table 3 are taken from a war climatology (World War I), when it was necessary to give an effective comparison of summer temperatures in Central Europe (Vienna, Austria: 48.2N, 16.3E, 660 feet) and the southern Adriatic coast (Scutari: 42.1 N, 19.4E, 70 feet). One might think that a simple comparison of temperature maxima would be illustrative. In reality, in Scutari, a maximum of 98F has been observed within a period of more than 30 years, in Vienna 98F was also observed, though within a period of 70 years. The maxima are practically identical. A frequency distribution of terryperature in Vienna and Scutari, however, shows immediately the characteristic and great climatic differences of the two places in spite of the fact that the comparison considers the temperatures only at 7 A.M. of the summer months of a synchronous period of 7 years. Each of the samples in Table 3 comprises 92 X 7 = 644 observations. TABLE 3. DISTRIBUTION OF RELATIVE FREQUENCIES (%) OF TEMPERATURE AT 7 A.M. AT A PLACE IN CENTRAL EUROPE (V) AND AT A PLACE ON THE SOUTHERN ADRIATIC COAST (S) IN SUMMER (JUNE, JULY, AUGUST) Class interval V S F % % 41.0 to 49 .9 0.5 _ 50.0 to 58.9 35.4 59.0 to 67.9 59.2 19.9 68.0 to 76.9 4.9 703 77.0 to 85.9. . 98 S 100.0 100.0 From Table 3 it is seen that at 7 A.M., after the cooling during the night, 95% of all observations are below 68F in Central Europe; on the Adriatic coast 80% exceed 68F. At first glance, GENERAL STATISTICAL CHARACTERISTICS 21 it is also seen that the distribution is less scattered in Scutari (3 classes) and more in Vienna (4 classes). These facts reveal a drastic contrast of temperature conditions in the two regions. It is not the absolute height of the maxima which counts. It is, rather, the fact that the greatest frequency is shifted to the next higher class on the Adriatic Sea. That means great frequency of high temperatures during the night and in the morning hours, a condition enervating for people accustomed to a more moderate climate. The frequency distribution points to the greater monotony of high temperatures in the regions of the Adriatic Sea during the summer. The foregoing example shows the value of relative frequencies for purposes of comparison. Besides, the size of the class interval is notable. With the sample in Table 3 the size of the interval is not less than 9 F, sufficient for the desired purpose. II. 5. HISTOGRAM AND FREQUENCY CURVE. THE MODE. THE MODAL CLASS As a consequence of the relatively small intervals in Table 2, the distribution takes a rather irregular shape, which is well illus- trated by the histogram of Figure 4. This figure contains two methods of representation: the histogram, " rectangular areas standing on each grouping interval showing the frequency of observations in that interval," 2 and the frequency curve which has been smoothed. In reality, the variation in question is dis- continuous, because the variate is confined to whole degrees Fahrenheit (see Table 2). The frequency curve is more usual in climatological statistics, and there is no real reason "for insisting on the histogram form." The latter may be more appropriate from a theoretical standpoint. The continuous curve gives a better survey, and the maximum of the curve indicates the approximate value of the mode, the most frequent value. This procedure is based on the assumption that a large number of observations are available to give a smooth distribution of frequencies. In the present example, that is not realized. 2 R. A. Fisher, Statistical Methods for Research Workers (6th ed., London, 1936), p.38. The frequencies can be plotted on the ordinates through the midpoints of the class in- tervals. "The tops of all the ordinates are joined by a broken line, 11 i.e. the original frequency curve which appears as a smoothed line in Figure 4. See D. Brunt, The Combi- nation of Observations (Cambridge, England, 1931), p. 7. 22 METHODS IN CLIMATOLOGY Another important concept is the modal class, that is, the class interval into which the greatest number of observations fall. In the example of Table 2, there are two equal modal class intervals, nos. 9 and 10. This offers a certain difficulty in O> 10 K *> 0> fOCSICMM -- ii i | I ' l IO t- lO <J> lO 04 8 - FIG. 4. Histogram of relative frequencies. A smoothed frequency curve is added. Minimum temperatures of four months of January at Mount Wash- ington, N. H. puting the temperature which represents the mode. Even in this case, histogram and frequency curve permit an estimate of the mode. In the example in question, the mode = ( 2.5 + l.S)/2 = - 0.5F. Generally, the value of the mode should be approximated by a simple linear interpolation ; the equation reads : 8 Mode = L mo + C H. Arkin, R. R. Colton, An Outline of Statistical Methods (4th cd. f New York, 1939), p. 23. GENERAL STATISTICAL CHARACTERISTICS 23 Where L mo = Lower limit of modal class f a = Frequency of class interval above modal group /6 = Frequency of class interval below modal group C = Size of class interval As stated before, in the example of Table 2 the class intervals nos. 9 and 10 include equal numbers of frequency. Therefore they are combined into one modal class with a frequency num- ber = 38. In order to avoid unequal intervals, nos. 11 and 12 are taken collectively as/ a and nos. 7 and 8 as/6. Thus we get from Table 2 : Lmo = 4F (lower limit of interval no. 9) f a = 13 + 9 = 22 (nos. 11 and 12) / 6 = 13 + 16 = 29 (nos. 8 and 7) C = 8F Therefore : Mode = - 4 + 22^29 8 = ~ aS5 F Accidentally, the two results are in perfect agreement. Generally, frequency distributions are necessary and indis- pensable in climatological investigations. This does not hold to the same extent where mode is considered. Variates taken from the realm of climatology, in particular, show remarkable frequency curves with two or more modes, so that the mode can no longer be considered as an unequivocal characteristic. Frequency curves of winter temperatures, for instance, show this phenomenon, especially at places located in a transition zone between an oceanic climate and a strongly continental one. Two modes exhibit apparently just about the same frequency: one, at a temperature of some degrees below the freezing point, corresponding to the average temperature of fresh continental air masses (Leningrad, 59.9N, 30.3E: about 20F); the other mode, at about the freezing point. This means that the heat content of the advected oceanic air masses at higher temperatures is expended in the process of melting the snow-cover, its thermostatic effect. We learn from this example that, in the great number of similar cases, two (and even more) modes can exist, so that the bulk of 24 METHODS IN CLIMATOLOGY observations cannot be characterized by one number alone, in contradistinction to the arithmetical mean. In other respects, also, the mode of climatic series happens to have properties which do not make it a desirable statistical element. On the other hand, one sees from the last example that the double mode of the tem- perature at a place (like Leningrad on the Baltic Sea) leads to a clear physical analysis of an important climatic feature. Even in this respect, it should not be forgotten that the frequency curve conveys the same knowledge, and moreover a comprehensive view of all observations. II. 6. ELEMENTARY CHARACTERISTICS. MEDIAN, QUARTILES, DECILES The method of computing the median stands between the primitive and the elementary characteristics, because there are at least theoretically two methods, the first of which leads immedi- ately to a primitive element. All items of which the sample consists are arranged according to size. This series is called the array. If the number of these items is odd, then the middle valu^ is called the median. If the number is even, then the average of the two central items is de- fined as the median. This method obviously represents the median as a primitive characteristic. Example: Take the series of minimum temperatures in January 1938 from Table 1 and arrange them according to magnitude. The series of Table 4 A results. The median = 2F. If the last TABLE 4. A. EXAMPLE FOR AN ARRAY. (TEMPERATURE MINIMA JANUARY 1938, FROM TABLE 1) No. 1 2 3 4 5 6 7 8 9 10 11 -15 -13 -13 -13 -11 -10 -10 -6 -5 -5 -5 No. 12 13 14 15 16 (median) 17 18 19 20 -4 -4-4-3-20111 No. 21 22 23 24 25 26 27 28 29 30 31 2 2 4 4 5 8 8 9 11 15 19 B. GROUPED DATA 4 No. F Frequency SF< 1 -15 to -7 7 7 2 - 6 to 2 15 22 3 3 to 11 7 29 4 12 to 20 2 31 GENERAL STATISTICAL CHARACTERISTICS 25 item (no. 31 = 19) were omitted so that the series had an even number of items = 30, the median would be the average of the two central items, nos. 15 and 16: - - 2.5F This method of arranging is " primitive, M but as a rule very tedious, because of the magnitude of climatological samples. The second method must be applied to samples which are al- ready grouped into class intervals. For this purpose, a sample with a greater number of items is necessary. Table 5 contains a sample TABLE 5. FREQUENCY DISTRIBUTION OF MINIMUM TEMPERATURE (C) OF 25 MONTHS OF JULY* (Coordinates for the place K: 48.1N t 14.1E, 1280 feet) 0) (2) No. (0 Central height C (3) Ft (4) -u 2 Fi -i 1 6.45 1 1 2 7.45 1 2 3 8.45 4 6 4 9.45 15 21 5 10.45 52 73 6 11.45 84 157 7 12.45 131 288 8 13.45 121 409 9 14.45 108 517 10 15.45 114 631 11 16.45 75 706 12 17.45 45 751 13 18.45 13 764 14 19.45 9 773 15 20.45 2 775 * Numbers of Table 5 taken from F. Steinhauser, Meteorologische Zeitschrift, 1935, p. 207. of daily minimum temperatures (C) for 25 months (of July at a place in the foothills of the eastern Alps). The absolute extremes of this sample are: 6.3C and 20.9C The range of variation is: 14.6 C The number of items is: 775 As this is a sample of greater magnitude with an exactness of 26 METHODS IN CLIMATOLOGY 0.1 C of the individual observations, 1 C is chosen as a suitable size for the class intervals. The first interval reaches from 6.0C to 6.9C the second from 7.0C to 7.9C the last from 20.0C to 20.9C The central heights of the class intervals are therefore : The first column of Table 5 contains the number of the class interval; the second, the central height of the class interval; the third, the frequency within the class intervals; and the fourth, the series of cumulated frequencies. i=15 This series ^i is calculated as follows : The frequency of the t=i interval no. 1 = 1; the frequency of nos. 1 and 2 = 1 + 1 = 2 etc. The last sum (at no. IS) represents therefore the whole size of the sample, i.e., 775. 4 The median value is defined in such a way that 50% of the sample lies below it and 50% above. The whole size is: N = 775 50% otN: f = 387.5 // From column 4, it is seen that this limit is surpassed within the 8th class interval. Therefore the median is located between 1 3 .0C and 13.9C. In this case, a simple linear interpolation leads also to the result. The 8th interval contains 121 items, of which 99.5 (387.5 - 288) should lie below the median value. Consequently, the interval of 1.0C (if we start from 12.9C) has to be divided in the propor- 4 S, the capital Greek S, means the sum of a given number of consecutive elements of a series. b 2 means that the sum is to be taken from the element with the number (subscript) a a to an element no. b. GENERAL STATISTICAL CHARACTERISTICS 27 tion of 99 5 - - 82 so that the median is at 12.9C + 1.0 X 0.82 = 13.72C. 6 According to the same principle, the entire size of the sample can be divided into an arbitrary number of equal parts, which are symmetrical about the median. It is usual to divide the sample into 10, or into 4, parts. The limits of the first are called deciles, those of the second, quar tiles. In the example of Table 5, the limits of the deciles are given by the cumulative sums of N/W = 77.5, with the following results: Decile no. 12345 Upper limit number of items 77.5 155.0 232.5 310.0 387.5 Decile no. 6 7 8 9 10 Upper limit number of items 465.0 542.5 620.0 697.5 775.0 The procedure of computation is now identical with that used for the median. The first decile contains 77.5 items, and therefore it lies within the 6th class interval between 11.0C and 11.9C. Because the first 5 class intervals contain 73 items, the 6th class interval participates with 4.5 items to give the first decile. There- fore, the entire size of the 6th class interval has to be divided in the proportion of g - <* The upper limit of the first decile is consequently at: 10.90 + 1.0 X 0.05 = 10.95C Herewith it is assumed that the sample is composed of discrete numbers, e.g., 6.0, 6.1, etc. 6 Part B of Table 4 shows the data of part A grouped into class intervals of 9F. There -15.5 Because the first interval contains 7 items, 8.5 items (15.5 7 = 8.5) of the second interval lie below the median. Therefore this interval of 9 F with 15 items must be divided in the proportion of and the median lies at: - 7 + 9 X 0.57 = - 2F. It is worth noting how good the agreement is between the method of an array (see Table 4A: 2F) and the method in question, even for a sample of only 31 elements. 28 METHODS IN CLIMATOLOGY Reduced to one decimal place, we get 1LOC The second and first deciles together contain 155.0 items. The second decile also lies within the 6th class interval which participates with 82 items. The class interval must be divided in the proportion of: And the upper temperature of the second decile lies at : 10.9 + 1.0 X 1.0 = 11.9Cetc. Thus the following division of the sample results : C Absolute minimum 6.3 Decile 1 11.0 Decile 2 11.9 Decile 3 12.5 Decile 4 13.1 Decile 5 * 13.7 = median Decile 6 14.4 Decile 7 15.1 Decile 8 15.8 Decile 9 16.8 Absolute maximum 20.9 The deciles divide the sample into equal parts of 10%; the quartiles into parts of 25%. As stated above, there is no other difference in principle between them. The computation of the quartiles needs no further explanation. In the example of Table 5, N = 775, the cumulative sums for the quartiles would be 193.75; 387.50; 581.25; 775.00. Therefore: Absolute Lower Upper Absolute minimum quartile Median quartile maximum 6.3 12.2 13.7 15.5 20.9C 5.9 C 1.5C 1.8C 5.4 C It is characteristic of the element under investigation that 50% of all observations lies within 3.3 C (between 12.2C and 1S.SC). The other 50% is distributed over a range of 11.3C altogether. These facts give a desirable picture of the rarity of greater devia- tions from the mean value. GENERAL STATISTICAL CHARACTERISTICS 29 The size of the interval between upper and lower quartiles is always of interest because 50% of the entire variate falls within these limits and 50% beyond them. There is the equal likelihood that an observation falls inside or outside of the limits of the quartiles. The small range between the lower and upper quartiles shows that the density of the items is much greater inside this space than outside. The three characteristics of the frequency curve median, quarries, deciles have been discussed, and a simple method has been shown of computing these statistical values. This has been necessary because in the English meteorological and climatological literature these characteristics are frequently used. Furthermore, these methods assume their real value only if the size of the sample is so large that half the number of its items represents a series of many hundreds. If, then, the comparison of the characteristics in question showed significant differences between two or more periods of years, we should infer "that the climate (or the methods of measuring it) had materially altered." 6 On the other hand, incontestable climatic series of great length large samples are fare; with small samples, the results from these characteristics are often disappointing. Further discussion of this subject is therefore superfluous. The frequency curve it- self is undoubtedly one of the best and finest means in the hand of the climatologist. R. A. Fisher, Statistical Methods, p. 42. CHAPTER III SPECIAL STATISTICAL CHARACTERISTICS OF CLIMATOLOGICAL SERIES III. 1. THE ARITHMETICAL MEAN IT GOES without saying that the arithmetical mean is defined by: a = a where a denotes the arithmetical mean, a,i represents any number from a\ to a n and n is the number of observations which constitute the series. The computation of the arithmetical mean even of large series is simple, especially if a calculating machine is at hand. If the size of the sample is extraordinarily large, there are methods for com- puting the arithmetical mean from* the sample divided into class intervals, when the midpoints and the frequencies are known. For these, the reader is referred to textbooks of statistical methods, since such cases are not sufficiently significant for climatology. The use of the arithmetical mean (average) in climatology is common and need not be fully discussed. Here, however, a few remarks about averages of extremes may be opportune, because of frequent misunderstandings. The following examples are taken from Table 1, minimum January temperatures on Mount Washing- ton, New Hampshire. (a) Mean daily extremes; in this case "minima" 1938: -1.1; 1939: -4.7; 1940: -8.7; 1941: -1.5F (b) The x-years' average of the mean daily extremes ; in this case the four-years' average of the mean daily min- ima = - 4F (c) Average absolute extremes calculated from the absolute extremes of each year (month etc.) ; in this case lowest minima: (-15, -36, -30, -30): 4 = - 28F highest minima: ( + 19, +20, +11, +21):4 = + 18F (d) The x-years' average range of the extremes, in this case of the minima, is 46 F 30 SPECIAL STATISTICAL CHARACTERISTICS 31 The examples are taken arbitrarily from the topic of temperature in order to dispense with additional tables ; the underlying concepts are applicable to any element of scalar character. III. 2. DEVIATIONS. AVERAGE VARIABILITY. STANDARD DEVIATION. NORMAL DISTRIBUTION There may be a series of n numbers representing the results of n observations : 0i, #2, #n The arithmetical mean is: Then the differences between the single elements of the series a f - and the arithmetical mean, i.e., di = a\ a; d% = a% a d n = a n a are called deviations, which play the greatest role in the representa- tion of statistical and also of climatological series of numbers. The sum of the deviations from the arithmetical mean is zero, if the signs of the deviations are considered. 1 The sum of the deviations, without regard to sign, divided by their number is called mean deviation (/z) in statistical mathematics and average variability (AV) in climatology. Therefore : n Here: S| | means the sum of the absolute values, ignoring signs: M = (\di\ + \d*\ + + \d n \):n Table 6 gives examples for the computation of deviations and of the average variability (mean deviation). The dates of the last killing frost in Vancouver, B.C., and in Bismarck, N. D., in the 1 In American papers and books the expression "deviation" is frequently replaced by the term "departure" and "mean deviation" by "mean departure." The term "departure" is seemingly not so common in British investigations and is even lacking in the Meteorological Glossary (London, 1939), and in Shaw's Manual of Meteorology. The term "average variability" is usual in numerous climatological papers of many countries and has its advantages from the didactic standpoint. 32 METHODS IN CLIMATOLOGY TABLE 6. DATES OF THE LAST KILLING FROST IN VANCOUVER, B.C., AND IN BISMARCK, N.D., AND DEVIATIONS (d) FROM THE MEAN DATE (rft) Year Vancouver, B. C. Date Year day* (D (2) d (3) d* (4) Date (5) Bismarck, Year day* (6) N. D. d (7) <** (8) 1901 IV 24 114 + 6 36 VI 7 158 +24 576 2 12 102 - 6 36 IV 29 119 -15 225 3 15 105 - 3 9 V 5 125 - 9 81 4 23 113 + 5 25 14 134 5 III 31 90 -18 324 12 132 - 2 4 6 IV 12 102 - 6 36 27 147 . +13 169 7 29 119 + 11 121 14 134 8 28 118 + 10 100 2 122 -12 144 9 21 111 + 3 9 13 133 - 1 1 10 14 104 - 4 16 17 137 + 3 9 +35 +40 2 1078 -37 712 1341 -39 1209 72 79 in IV 18 108 7.2 V 14 134 7.9 a 8.44 11.00 * Number of the day from first of January. The data are taken from Frank J. Bavendick, "Climate of North Dakota," p. 1048, and Lawrence C. Fisher, " Climate of Washington, " p. 1175, Climate and Man, Yearbook of Agriculture (Washington, D. C., 1941). ten years 1901 to 1910 are given. For computation, the dates (columns 1 and 5) have to be converted to numbers of days starting January 1. (February 1 = no. 32, etc.; see below, Appendix IV). These numbers are shown in columns 2 and 6. The average dates (arithmetical means) are the 108th and 134th days of the year, or April 18 and May 14 respectively. In columns 3 and 7, the devia- tions are shown and their sum total (disregarding the signs) is given. The sums of the positive and of the negative deviations should be identical (small differences being due to rounding off the arithmetical mean (rri) to integers). 2 The sum of the absolute values of the deviations divided by their number (10) equals the average variability, which represents one of the most valuable climatological elements derived from the observations. From the examples, the variability of the date of the last killing frost is somewhat greater in the interior of the continent than at the coast. The principal contrast is given by the differ- 2 The differences should be checked. In the case of column 2, the exact rh = 107.8. Each of the 10 deviations is afflicted therefore with an error of 0.2. Hence, the differ- ence 10 X 0.2 = 2, in agreement with the result in column 3. In column 6 the exact mean is 134.1 instead of 134. The difference is 0.1 X 10 1. SPECIAL STATISTICAL CHARACTERISTICS 33 ence in the average dates. Inland, the average last killing frost occurs 26 days later than at the coast. A rough measure of the variability is also given by the difference between the absolute extremes, i.e., 29 days at Vancouver and 39 days at Bismarck, N. D. This measure is called "rough" because it depends upon only single values. For instance, if the series began with 1902, the differences would have been 29 days (un- changed) and 28 days, respectively. The average variability considers each value of the series, and is the usual method of characterizing the scatter of the element in question. The second method is to compute the standard deviation 8 (a) cr = which is the best and most exact measure of scattering. In climatological investigations it is not so frequently used, since the calculation is more complicated. This problem will be treated later. The example of Table 6 gives the following results (columns 4 and 8): = 712 2<Z 2 * = 1209 The standard deviation, the scatter of the date of the last killing frost, is about 30% greater inland than at the coast. From the frequency distribution shown in Figure 4 and Tables 4B and 5, one draws the simple and natural conclusion that ex- treme values are rarer than values close to or within the modal class interval. Under certain conditions, this experience holds, too, in the case of deviations if the inferences are logically changed. This means that the smaller the deviations, the more frequent they are. The ideal case is realized by the so-called normal distribution, or Gaussian distribution, which governs deviations independent of one another. This is the " standard deviation" of the single value; that of the arithmetical mean is: ' _L * Vn 34 METHODS IN CLIMATOLOGY There should be no effective reason why the deviations, if their number be large, may not take any value from <*> to + Then, the frequency curve is a bell-shaped curve which fulfills the equation : y = where y means the frequency ordinate at the distance d (deviation) from the center of the curve (Fig. 5), TT = 3.142 (Ludolf s number), FIG. 5. Normal distribution showing the areas (per cent) between ordi- nates in distances of one, two, three Standard Deviations left and right from the center (frequency of the arithmetical mean) e = 2.718 (base of the natural logarithms). This is the law of Karl Friedrich Gauss. 4 Obviously, the curve represented by this equation is sym- metrical, with the greatest frequency at the center, because equal values of y are related to ( d) and to (+d). The greatest fre- 4 For a more correct representation, L. H. C. Tippett (The Methods of Statistics, London, 2nd ed. f p. 54) may be quoted: "The frequency curve is deduced from a histo- gram and consequently areas under the curve, not heights of the ordinates, represent frequencies. It is therefore appropriate to use the notation and ideas of the integral calculus, to imagine about any given value of x an elemental sub-range dx, and to regard the area under the curve between ordinates drawn at the limits of the sub-range as an element of frequency, df. Then this elemental strip may be regarded as a rectangle of N height y and df = = e^^'dx" where N denotes the number of items. SPECIAL STATISTICAL CHARACTERISTICS 35 quency occurs at d = 0, if Therefore the arithmetical mean is the most frequent value, and coincides with the values of the median and the mode. This is valid only if there is a symmetrical distribution of frequency of the deviations with an absolute maximum at d = 0. Gauss's equation shows that a normal frequency distribution is fully defined by its respective standard deviation <r. Geometrically, <r is the distance on either side of the center where the slope is steepest, at the points of inflexion of the curve. (See Fig. 5.) In practical applications we are not so much interested in the frequency at any distance from the center. It is much more im- portant to know the total frequency (or probability) beyond this point (for instance beyond the distance a in Fig. 5). This fre- quency or probability is represented by the area included between the ordinate at the chosen point, the tail of the curve, and the X-axis. For deviations expressed in units of cr, (d/d), tables of this total frequency, or probability integral, have been constructed, from which, for any value d/cr, we can find what fraction of the total sample has larger deviations. III. 3. A PRACTICAL APPLICATION OF THE STANDARD DEVIATION IN CLIMATOLOGY The probability (P) 5 that a deviation exceeds 1, 2, or 3 times the standard deviation is 0.317; 0.047; 0.003. The numbers illustrate the rapid decrease of the probability of increasing deviations. In other words, the area between the ordinates at: cr (Fig. 5) 2cr, 3cr includes: between - or and -f a: 68.260% between -la and +2cr: 95.346% between -3<r and +3cr: 99.729% between -4(7 and +4cr: 99.994% 1 That is the ratio of the number of favorable events to the total number of possible events. If a coin is tossed, only " heads" or "tails" can appear. The probability of heads (e.g.) is therefore 1/2. 36 METHODS IN CLIMATOLOGY In climatological investigations, the probable error (f) can be useful because it includes exactly 50% of the sample between +/ and /. The probability that a deviation is </ is therefore 1/2. The distance of the ordinate from the center which borders this area of the curve is called quartile distance^ a term the meaning of which is clear from previous explanations. The percentages of the areas included between the ordinates at /, 2/, etc. are: between + / and - /: 50.00% between +2/and -2/: 82.26% between + 3/and -3/: 95.70% between +4/and -4/: 99.30% The probable error is usually computed from the standard deviation according to the relation : / = 0.6745<r Example of the practical use of the division of the area of the curve of normal distribution Often the climatologist has to determine the rareness of an actual observation of temperature, precipitation, etc., or else of a mean intensity, or amount, such as a mean of temperature, sum of precipitation of an individual month, etc. Statements to that effect are necessary on the occasion of an expert opinion and in the description of a climate. Questions may arise such as: Is a wind velocity of x mi/hour, is a temperature of x degrees below zero, is a downpour of x inches in an hour, etc., a normal event in a given climatic region, or is the observed value rare, or very rare? The courts are especially interested in such estimates because in the latter case the judge can declare the event to be a force majeure (superior force). For many other purposes, including military ones, it is highly important to know the limits between which normal values of the climatic elements can vary. Usually such questions are answered by the climatologist ac- cording to common sense. The division of the area of the curve of normal frequency distribution provides a transition from guessing to quantitative estimation. If a long series of observations of the element in question is at hand, one can calculate cr or/ (standard deviation or probable error). SPECIAL STATISTICAL CHARACTERISTICS 37 Then the following nomenclature can be used : below: 3<r( 3/): extremely subnormal = ES between: 3o-( 3/) and 2<r( 2/): greatly subnormal 2o-( 2/) and <r( /): subnormal = 5 ~ *( f) and + cr(+ /): normal = J + <jr( + /) and +2cr( + 2/): above normal = A + 2cr( + 2/) and + 3 cr ( + 3/) : greatly above normal = above: +3o-( + 3/) extremely above normal = JS-4 At State College, Pennsylvania, 6 53 years' observations of the mean daily minimum of the coldest month were available. The probable error is:/ = 2.9 F. Hence the following classification (as indicated in Table 7). The class limits are derived from the 53 years 1886 to 1938. It should be kept in mind that these limits are changed if another period is used. On the other hand, the variability of the limit is smaller by far than that of the values of the element itself. In climatographies, such classifications of temperature, precipitation, etc., should be given for each place with about 50 or more years' observations. From Table 7, one concludes that a deviation from the mean daily minimum of > 8.7F is to be considered a rare event, with a theoretical probability of two occurrences in 100 years. Conse- quences from such low temperatures (ruin of winter wheat, of rosebushes, etc.) could be judged as caused by a " superior force/ 1 TABLE 7. CLASSIFICATION OF DEVIATIONS FROM THE MEAN DAILY MINIMUM TEMPERATURE AT STATE COLLEGE, PA., BASED ON THE PROBABLE ERROR (1) (2) (3) (4) (5) Charac- Observed Calculated teristics Classification frequency frequency Obs. - Calc.* ES < - 8.7 F 1.88 1.80 +0.08 GS -8.7 to -5.8 7.55 6.72 +0.83 S -5.8 to -2.9 18.87 16.13 +2.74 N -2.9 to +2.9 45.28 50.00 -4.72 A + 2.9 to +5.8 18.87 16.13 +2.74 GA + 5.8 to +8.7 7.55 6.72 +0.83 EA > +8.7 0.00 1.80 -1.80 * Column 3 minus column 4. 8 V. Conrad, "Investigation into Periodicity of the Annual Range of Air Tempera- ture at State College, Pennsylvania," The Pennsylvania State College Studies, No. 8 (State College, 1940), pp. 10 ff. See also F. H. Chapman, Meteorological Office, London: Professional Notes, 1919, no. 5; and V. Conrad, Meteorologische Zeitschrift, 1921, p. 91. 38 METHODS IN CLIMATOLOGY The example in Table 7 shows that the " probable error" has the advantage that the intervals are rather smaller than at the standard deviation. This is valuable for climatical purposes since limits of rare cases are better adjusted to common feeling. The theoretical frequency distribution is represented as a full line in Figure 6, and the crosses mark the observed frequencies at the midpoints of the class intervals. It should be pointed out that o/ 7 20 10 -10.2 -7.3 -4.4 -1.5 1.5 4.4 7.3 !0.2F FIG. 6. Frequency distribution (per cent) of average daily minimum temperatures in State College, Pa. Calculated : Full line. Observ%d : x.* (After V. Conrad) % * The numbers below the abscissa mean the midpoints of the class intervals. The normal class is divided into two class intervals. the observed frequency curve fits the normal distribution very well. The small departure within the modal class interval is called positive excess. III. 4. VARIANCE. COEFFICIENT OF VARIATION Before continuing the discussion of how far climatological series show a more or less normal frequency distribution, one may men- tion two further measures of the dispersion (scatter) of a variate. Both, e.g., are used in the papers and reports of the agricultural climatological department of British India. 7 They are directly derived from the standard deviation (o-). The first is the variance ( Va) : Va = n 1 See, for instance, P. C. Mahalanobis, Report on Rainfall and Floods in North Bengal (Calcutta, 1927). SPECIAL STATISTICAL CHARACTERISTICS 39 It is the square of the standard deviation, and is somewhat easier to calculate. The indications of the variance react a little more strongly to small changes of the distribution of deviations, as exemplified in the left side of Table 6 (Vancouver) : . n 10 The other measure is the coefficient of variation (CV), that is, the standard deviation expressed as a percentage of the arithmetic mean (m) : CV= 100^=100 if the numerical values are taken from Table 6 (Vancouver). The "coefficient of variation" should be more frequently used in climatological statistics; it permits an exact comparison of the values characterizing the dispersion because it is reduced to an equal arithmetical mean. III. 5. SUMMARY OF STATISTICAL CONCEPTS Thus far the following statistical expedients have been ex- plained : Frequency Distribution Mean Deviation (/*) Median Standard Deviation (<r) Quartiles Normal Distribution Deciles 8 Probable Error (/) Arithmetic Mean (in) Variance (Va) Deviations (d) Coefficient of Variation (CV} III. 6. RELATION BETWEEN /x, <r, AND /. CORNU'S THEOREM. UNILATERALLY AND BILATERALLY LIMITED VARIATES In the foregoing discussion only the relation of standard devia- tion (<r) and probable error (/) was mentioned : / = 0.67450- Standard deviation (<r) and mean deviation (/z) are connected by 8 To my knowledge, sextiles (6 parts) and percentiks (100 parts of a sample) are not usual in climatological investigations. For the rest, the method of calculation is iden- tical to that used for medians, quartiles, etc. 40 METHODS IN CLIMATOLOGY the following simple equation : <r = A/^ ji = 1.253/i As a rough approximation one can use a = 5/4 M These relations are exactly valid only in the case of a normal distribution, and approximately valid if the departures from the normal distribution are not too great. If the size of the climato- logical sample is not too small (lower limit perhaps 25 items), a can be computed with the above equation if there is some likeli- hood that the variate can be represented more or less by a normal frequency distribution: An example may be taken from Table 6. The mean deviation /* (column 3) is: M = 7.2 therefore , = 1.253/i = 9.02 instead of 8.44. This means an error of 7% of the true value. The agreement mentioned is not too good, because the number of items is much too small. The equation a = can be written ^ = TT = 3.1416 This expression is called Cornu's Theorem and has often been used as a solution of the problem, whether or not one may assume that the climatological elements can be represented by a normal distribution. This inversion of the problem may be illustrated by the following examples. The deviations of 125 mean winter temperatures at Vienna, Austria (Central Europe), yield - 3.H M 2 125 mean summer temperatures at Vienna, Austria, yield - 3.25 SPECIAL STATISTICAL CHARACTERISTICS 41 130 average annual temperatures at Paris, France (Western Europe), yield = 3.16 M 110 average annual temperatures at Milan, Italy, yield ^ = 3.11 M 2 130 average January air pressures at Paris yield ^-3.13 M 2 From the ten years' data of the last killing frost in Vancouver (Table 6) it follows that: 2<r 2 X 8.44* The differences between the other results are only 4% of TT. One might think that in general the series of deviations of a climatological element shows the normal distribution. An ex- ample of this behavior is given in Table 7 and Figure 6. On the other hand, it must be emphasized that the fulfilling of Cornu's theorem is only a necessary, but not a sufficient condition of normal distribution. This means that if Cornu's criterion is not satisfied, no normal distribution can be expected; but the criterion can be fulfilled even if the deviations obey another law of distribution. Therefore the inferences from Cornu's theorem are not binding. Finally, one point, often overlooked in textbooks and in scien- tific papers, must be stressed. A normal distribution is possible only if there is as stated earlier no effective reason that the deviations should not take all values from <*> to + This requirement is more or less fulfilled in the case of pressure, of air temperature, and of some derived elements. Many elements have zero as the limit on one side; e.g. precipi- tation, wind velocity, etc. In spite of this fact a more or less normal frequency distribution and inferences from it are not excluded if the frequency curve shows only relatively small frequencies in the vicinity of the given limit (e.g., zero). 9 9 Naturally also skewness and kurtosis (peakedness, excess) have to be considered besides dispersion if an exact statement is to be made as to whether or not there is a Gaussian distribution. See the rain example in R. A. Fisher, Statistical Methods, p. 54 ff. 42 METHODS IN CLIMATOLOGY The precipitation during March in Helwan, Egypt (Table 8), offers an example of an extreme case. The deviations yield : a = 8.03 mm The arithmetic mean, however, is 6.3 mm of rain in March on the average for 21 years. TABLE 8. RAIN IN MARCH AT HELWAN, EGYPT (29.9N, 3i.3E, 38 FT.) Year A Amounts (mm) B Deviations Squares 6f the deviations* 1904 -6.3 39.7 5 1.9 -4.4 19.4 6 4.8 -1.5 2.2 7 9.1 2.8 7.8 8 24.8 18.5 342.2 9 -6.3 39.7 10 7.8 1.5 2.2 11 0.4 5.9 34.8 12 -6.3 39.7 13 1.6 -4.7 22.1 14 -6.3 39.7 15 -6.3 39.7 16 25.8 19.5 380.2 17 > -6.3 39.7 18 13.0 6.7 44.9 19 -6.3 39.7 20 7.2 0.9 0.8 21 9.1 2.8 7.8 22 7.5 1.2 1.4 23 -6.3 39.7 24 19.4 13.1 171.6 2 133.9 1355.0 rh 6.3 C. Deviations Grouped into Class Intervals Class interval Midpoint, mm Frequency % -7.0 to 1.1 mm -4.0 57 -1.0 to 4.9 2.0 24 5.0 to 10.9 8.0 5 11.0 to 16.9 14.0 5 17.0 to 22.9 20.0 9 * Rounded off to the first decimal place. The negative deviations are crowded within the first class interval (Table 8, C) and the frequency curve has nothing to do with the normal distribution as seen in Figure 7. These conditions are exaggerated even more if the variate is bilaterally limited. SPECIAL STATISTICAL CHARACTERISTICS 43 -4.0 2.0 8.0 14.0 20.0 mm FIG. 7. Frequency curve of rainfall in March (21 years) at Helwan, Egypt. On the X-axis: Midpoints of the class intervals of the deviations (mm) FIG. 8. Frequency distribution (per cent) of the degrees of cloudiness in ten September months at 9 p.m., at Prague, Czechoslovakia 44 METHODS IN CLIMATOLOGY Perhaps the best example is the element cloudiness with its two limits : zero for cloudless sky, and ten for overcast sky. Figure 8 shows the frequency distribution of cloudiness (in per cent) for ten months of September in Prague (50.1N, 14.4E, 663 ft.), at 9 P.M. The extremes are by far the most frequent values in contradistinction to the normal distribution, because of the bilateral limitation of the variate and of regional climatic proper- ties. 10 The arithmetic mean lies at about cloudiness 5, The arithmetic mean here loses its regular sense and indicates more or less the degree of mixing of the extreme values. To go back for a moment to the example of the March rain in Helwan: Cornu's theorem can be proved by means of the data given in Table 8. We get: or a value only 1% greater than Ludolfs number. This is a typical example of a series of an element with a frequency distribu- tion different on principle from the normal distribution. Never- theless, Cornu's theorem is nearly fulfilled. We repeat that it is a necessary and not a sufficient condftipn. III. 7. HIGHER CHARACTERISTICS (SKEWNESS) The higher characteristics are used to analyze frequency dis- tributions which are not identical with the normal distribution. For climatological purposes the asymmetry of the frequency curves is of interest. The term for it is skewness. In contradistinction to the symmetrical normal distribution, here arithmetic mean and mode no longer coincide. The arithmetic mean is much affected by extreme values of the variate ; the mode is not affected at all by the extremes. Therefore mean and mode cannot coincide. The greater the distance is between these two characteristics, the greater is the asymmetry, the skewness (Sk). According to what has been said before, the mean moves toward that side which shows the greater and more frequent extreme values. Therefore a frequency curve with the mean at the right side of the mode is called positively skewed (showing right skewness), while in the opposite case, it is negatively skewed (showing left skewness). Right and left skewness may be seen in Figures 9a and 9b. 10 The statistics are taken from L. W. Pollak, in Prager Geophysikalische Studien, vol. VI (Prague, 1931), p. 11. SPECIAL STATISTICAL CHARACTERISTICS 45 K. Pearson gives as a measure of skewness the difference be- tween mean and mode. This difference has to be reduced to equal dispersion. Thus the equation results: Sk = Mean Mode MODE MEAN FIG. 9a. A right (positively) skewed frequency distribution showing the theoretical position of mode and mean. (After Arkin and Colton) MEAN MODE FIG. 9b. The same as above, but left (negatively) skewed It is useful to know, for instance, how far temperature devia- tions of one sign predominate as to quantity and to frequency. The trouble is that the position of the mode is rarely calculated, because of the great labor involved for long series. 46 METHODS IN CLIMATOLOGY Less burdensome is Koppen's method of calculating the asym- metry (A). The equation reads: n where rib means the number of items below the mean, and n the total number of items. For the case of symmetry, A ~ since #6 = n/2. Example: C. F. Brooks gives the annual amounts of precipita- tion n for n = 80 years, at Philadelphia, Pa., expressed in devia- tions. The mean is added. A simple counting yields in a few minutes n b = 43 therefore A - 1 - - A " l 80 " Compared with other results, the negative asymmetry is rather high. Negative deviations are a little more frequent. The ex- pectation of a relatively dry yearns somewhat greater than that of a wet year. The great advantage of Koppen's method is that it can be com- puted easily, and the A from different places are comparable with one another. On the other hand, it should not be forgotten that Koppen's method considers only the frequency of the signs, and that the amounts of the deviations are ignored. This is the great disadvantage of this method compared with the skewness method. Other characteristics of the frequency distribution, such as the peakness of the curve of frequency distribution, moments, etc., are of less interest to climatologists, and are not discussed here. 12 III. 8. DIFFERENT KINDS OF VARIABILITY USED IN CLIMATOLOGY 1) Mention was made earlier of average variability, which gives a good measure of the variations to which the climatological ele- ment is subject. U R. De Courcy Ward and C. F. Brooks, "The Climates of North America," in Kflppen-Geiger Handbuch der Klimatologie, vol. II, pt, J, p. 283. 12 See R. A. Fisher, Statistical Methods; H. Arkin and R. R. Colton, An Outline of Statistical Methods (4th ed., New York, 1939); H. L. Rietz, Handbook of Mathematical Statistics (New York. 1924). SPECIAL STATISTICAL CHARACTERISTICS 47 The average temperature of January has an average variability ?\di\/n: 0.7 F in Batavia, Java (6.2S) 9.7 F in Green Harbor, Spitsbergen (78.0N) The average variability of the mean temperature of the months, is indeed climatically very characteristic. In the Arctic climate, the average variability of the temperature in January is about 14 times greater than that in the inner tropic belt. 2) If the series of observations is rather short (<10 years) the absolute range of variation of the series gives a first approximation of the variability of the element. 3) The average variability is independent of the sequence of items. This is, however, of special climatological interest. It is important in judging of a climate, for instance, whether the changes from one day to another are rather smooth or very abrupt. This method is not restricted to equal sections of time (e.g., the day). It can be applied also to equidistant places along a straight line, running, for instance, from the coast inland. Special variabilities, regarding the sequence of the places in a given direc- tion, could yield new characteristics of a special contrast in the climate of a region. Thus a measure of variability which considers the quantity as well as the sequence of the items of the series of numbers character- izing the variations of a climatic element is necessary. Table 9 offers an example of the "average variability f> (AV) computed from the series in part A: A V = 5.0. In part B another variability (SV) is calculated in the following way: SV = [|0i - 8 1 + 1 02 - as | + + kn-i - 0|]r (n - 1) if n equals the number of items. This variability may be called inter-sequential variability, because its value is determined by the quantity of the items as well as by their sequence. In the ex- ample, SV = 8.8. For the sake of clarity, the series is arranged as an array in part C of Table 9, to show how great the influence of the order of numbers is upon the quantity of SV. This is now a quarter of that in part B. This important method of calculating a variability which takes into account the sequence of the items in the climatological series 48 METHODS IN CLIMATOLOGY TABLE 9. COMPARISON BETWEEN AVERAGE VARIABILITY (AV) AND "INTER-SEQUENTIAL" VARIABILITY (SV) n * number of items, d = deviations from arithmetic mean (m). D =5 Difference between two consecutive items A Original order a\ oj a\ a\ a* a* &t a* a* aio of 10 numbers 9, 8, 17, 2, 9, 20, 0, 7, 12, 16; rh - 10 deviations -1, -2, +7, -8, -1, +10, -10, -3, +2, +6 S(J+) = 25, 2(d_) 25 B Computation of SF 1, -9, 15, -7, -11, 20, -7, -5, -4 from the series A Z(Z>+) - 36, S(Z>_) = - 43 Q n 1 9 C Series of A arranged 0, 2, 7, 8, 9, 9, 12, 16, 17, 20 as array Differences between 2, -5, 1, -1, 0, -3, -4, 1, -3 consecutive items >+) . 0, *(ZU - - 20 can be applied to each element as far as it represents a scalar quantity completely specified by one number, in contradistinction to directional quantities, which have direction as well as magnitude (wind, motion of clouds, motion of the water in an ocean cur- rent, etc.). The best-known example of an inter-sequential variability is the interdiurnal variability. This is calculated by two methods: 1) Differences of the consecutive daily means of the element; 2) Differences of the element at a fixed hour of consecutive days. The advantage of the first method is that, for instance, the mean daily temperature is derived from at least two observations (e.g., the extremes), so that incidental errors are not so likely. On the other hand, average values can always conceal interesting phenomena. The disadvantage of the second method is that, for instance, the interdiurnal variability of temperature has a well-accentuated SPECIAL STATISTICAL CHARACTERISTICS 49 daily variation. The greatest trouble is that the time of the ex- tremes is variable in the course of the year. In Central Europe (Potsdam), the maximum interdiurnal variability occurs at 6 A.M. in October, and at 4 P.M. in April. This daily variation reaches its maximum in July, when the ratio maximum/minimum is 1.8. Much caution should be exercised before drawing conclusions from investigations in interdiurnal variability taken from a fixed hour. Tables 10 gives a nearly complete example of an investigation in interdiurnal variability of temperature. 13 TABLE 10. DIFFERENT STATISTICS REGARDING INTERDIURNAL VARIABILITY OF TEMPERATURE Col. 2 = Interdiurnal Variability of Temperature (5 A.M.) (ID V) at Mount Washington, N. H. (6270 feet). Col. 3 = Average Coolings (C) Col. 4 = Average Warmings (W) Cols. 5 and 6 = Average Numbers of Consecutive Days with Increasing (WD) and Decreasing (CD) Temperature Col. 7 = Length of Temperature Surges in Days (S) (After V. Conrad) 0) Month (2) IDV F (3) C F (4) W F (5) CD days (6) WD days (7) 5 days Dec 8.7 9.2 8.3 1.84 1.92 3.76 Jan 11.4 13.8 9.6 1.44 1.86 3.30 Feb 10.1 9.9 10.4 2.36 2.45 4.81 Mar . . ..98 104 9 2 1.79 1.92 3.71 Apr. 62 66 5.9 1 62 2.13 3.75 May 5 9 63 5 6 1 79 2.13 3.92 June 5.1 5.2 5.0 1.79 1.95 3.74 July 4.0 4.4 3.7 1.74 2.13 3.87 Aug 4.2 4.4 4.1 2.05 2.40 4.45 Sept 6.3 7.5 5.2 1.87 2.13 4.00 Oct 8.0 8.3 7.7 2.09 2.23 4.32 Nov 8.0 8.5 7.5 1.96 2.04 4.00 (rft) 7.3 7.9 6.8 1.86 2.11 3.97 Column 2 contains the interdiurnal variability (IDV) at 5 A.M. The values are averages of 3 years. Irregularities of the annual course of ID V, as well as of the elements in the other columns, are to be explained by the shortness of the period under discussion. 18 V. Conrad, "The Interdiurnal Variability of Temperature on Mount Washing- ton," Transactions of the American Geophysical Union (Washington, 1942), pt. II, p.279. 50 METHODS IN CLIMATOLOGY The statistics which lead to the IDV (which ignores the signs) enable one to calculate separately the average amount of tempera- ture increasing (W) (col. 4), or decreasing (C) (col. 3) from day to day. The increase of temperature from one day to the next might be termed warming; the decrease, cooling. Because in the problem in question the amounts of cooling and of warming are not effectively influenced by radiation proc- esses, they are a measure, or at least an indication, of advected cold and warm air masses. Therefore, these statistics are valuable for climatology as well as for long-range forecasting, etc. When the IDV is calculated, average and maximum values of cooling and warming should be published. This can be done with little or no supplementary labor. A further valuable addition is represented by computing the number of consecutive days with increasing temperature (WD) (column 6), and with decreasing temperature (CD) (column 5). The individual numbers are averaged for each month. Column 7 shows the sums of CD + WD : that is, the average time in days for an average, consecutive increase and decrease. This seesaw can be called a temperature surge, and column 7 (S) gives the average duration of the surges in days. % Once more, it should be emphasized that the inter-sequential variability can and should be used when different climatological elements are discussed. For lack of space, only one example of temperature and the time-unit of one day is given in this chapter. For one purpose or another, an interhourly variability may be useful. As an example, the wind velocity (regardless of the direc- tion) could be studied from this standpoint, if the establishment of windmills in a given locality were planned. 14 B. M. Varney 15 uses an interannual variability to characterize the rain conditions of California. Annual and seasonal amounts are both handled by this method. It goes without saying that monthly average data for air pressure, temperature, cloudiness, vapor pressure, precipita- tion, depth of snow, etc., can be studied also by interannual, inter- seasonal, etc., variability. 14 See also V. Conrad, " Interhourly Variability of Temperature at Mount Washing- ton, N. H.," Transactions of the American Geophysical Union, 1943, p. 122. 15 B. M. Varney, "Seasonal Precipitation in California and its Variability," Monthly Weather Review, 1925, pp. 148-163, and 208-218. SPECIAL STATISTICAL CHARACTERISTICS 51 III. 9. ABSOLUTE AND RELATIVE VARIABILITY AND OTHER MEASURES OF VARIATIONS If amounts of such as precipitation are considered, the average variability ^ - - = AV depends naturally upon the arithmetic n mean The normal annual precipitation in Cairo, Egypt, is 1.34 in.; the variability is 0.67 in. This is large, because the annual pre- cipitation varies between one half and three halves of the mean. The same variation in Rangoon, Burma (17N,96E), with a 4 4 normal " precipitation of about 100 in., would be negligible indeed. Therefore, a new concept is necessary, no longer dependent on the arithmetic mean : the relative variability. This is the absolute variability expressed in per cent of the arithmetic mean. pup The relative variability V r would reduce the absolute variability AV to the unit of annual precipitation only if the correlation be- tween the two variants, V r and p, were linear. This assumption does not hold, as is shown from the observations. Statistics from about 360 stations scattered over the earth yielded a correlation between V r and normal rainfall. This is represented in Figure 10 by a hyperbolic curve. Therefore, the influence of the normal sum of precipitation becomes weak only beyond the turning-point of the curve, i.e., if the normal rainfall is larger than about 28 inches. Then, in a first rough approximation, V r is practically no longer dependent upon p. It should be stressed that about 40% of the earth's surface has less than 28 inches pre- cipitation, and 30% has less than 20 inches. Conclusions drawn from comparing values of V r for different places in these vast regions of small annual precipitation are inaccurate and mislead- ing. 17 A way out of these difficulties is offered by the representa- 16 If the arithmetic mean is calculated from a longer period, perhaps at least 25 years, it is called normal value or normal. The Meteorological Office in London pub- lished a Book of Normals based on a period of 35 years (see Xll.S.f). 17 For more details, see V. Conrad, "The Variability of Precipitation,' 1 MWR, vol. 69 (1941), pp. 5-11. Statistics taken from E. Biel, "Die Ver^nderlichkeit der Jahressumme des Niederschlags auf der Erde, M Geographisches Jahrbuch aus Oesterrcich, 14/15, 1929. 52 METHODS IN CLIMATOLOGY tion of anomalies, treated later (XIV, 7). In regions with an annual precipitation greater than about 20 to 28 inches, the values of V r can be compared with one another without serious error. Vo 90 SO 60 20 20 40 60 80 INCHES FIG. 10. Relative variability and rainfall. (After V. Conrad)* * v f in the diagram should read V r . Since the calculation of V r is rather troublesome for long series of observations in many investigations of precipitation conditions, a simpler measure of the variability is commonly used. G. Hell- mann introduced an expression which is called ratio of variation (Q) - 18 It is nothing but the quotient m where M = the largest annual amount in the series and m = the smallest. This measure should not be used, since Q becomes infinite in many places located in the desert belts. 18 G. Hellmann, Veroffentlichungen Preuss. Met. Inst. Ill (1909), no. 1. SPECIAL STATISTICAL CHARACTERISTICS 53 E. Gherzi used another measure of variation : 19 _ M m V p It is a great improvement over the measure M/m. III. 10. SOME APPLICATIONS OF THE METHOD OF RANDOM SAMPLES III. 10. a. Precipitation Though the following inferences are restricted to one or another element (precipitation and fog) similar methods could be applied to obtain results for other elements. At some thousands of climatological stations, eye observations are made from one to three times a day, but no continuous records are kept. One assumes that the observer enters the international sign for rain or snow in his register when it rains or snows at the precise time of the observation. Then, the observation can be valid for this time as a random sample of the actual weather. Starting from this idea, W. Koppen derived from these instantane- ous observations data which, otherwise, could be taken from con- tinuous records only. Probability has previously been defined as the ratio of the number of favorable or desired events to the total number of possible events. If one is interested in precipitation, observations, just at the moment of rain or snow, are the favorable events. In a given period (e.g., a month), n observations are made. Among these, r observations occur exactly during the precipitation, Then, the proper fraction r/n = p, is the absolute probability of precipitation; " absolute/' because independent of the selected unit of time. The absolute probability of precipitation is the basis of Koppen's estimates. The following statement and some simple inferences give a survey of the quantities in question and a demonstration of how to use them. n = Number of all observations within the chosen period (e.g., month, season) r = Number of all observations with precipitation 19 p = average amount of precipitation of the period in question. 80 E. Gherzi. tude sur la pluie en Chine (Observatoire Zi-Ka-Wei, 1928). 54 METHODS IN CLIMATOLOGY N = Total number of hours in the chosen period (e.g., 720 hours in month of June) d = Number of days with precipitation within the chosen period h = Amount of precipitation in the period in question y p = - = Absolute probability of precipitation n Then : D = pN = Probable total duration of precipitation in hours D pN , ... = j- = Average duration of precipitation, per day with precipitation, in hours h - = "intensity" of rain (average amount of rain per rainy d day) = h rr = Average amount of rain in one rainy hour D prJ Example: At the observatory in Batavia (Java), not only were all the necessary observations made by eye and by self-recording gauges, but also the data were published in full detail, so that the method described can be tested to see how far it is in agreement with observation. January 1915 21 was the month of the test (dry period). Regular observations were made three times a day: n = 93 By counting, it was determined that: r = 10 Therefore: p = r/n = 10/93 = 0.1075 Further: N = 31 X 24 =744 hours Hence : D = pN = 80 hours with rain Observed: d = 29 Therefore: D/d = 80/29 = 2.76 hours Observed: h = 13.07 inches Hence: h/d = 13.07/29 = 0.45 inches/rainy day and h/D = 13.07/80 = 0.163 inches/rainy hour From Table 11 it can be seen how well this method given by Koppen agrees with observation. The computation is simple and the method is easy to understand. On the other hand, it supplies information about data which as stated before can usually be 21 Yearbook of Batavia, Java. SPECIAL STATISTICAL CHARACTERISTICS 55 TABLE 11. METHOD OF RANDOM SAMPLES APPLIED TO RAINFALL AT BATAVIA, JAVA Number of Average duration of rain in Average amount of rain rainy hours one rainy day in one rainy hour D obs. D calc. D/d obs. hours D/d calc. hours h/D obs. in. h/D calc. in. 75.3 80 2.6 2.8 0.173 0.163 obtained only from continuous records. These ideas could be used, with the necessary modifications, for other climatological elements. III. 10. b. Cloudiness (C) and Duration of Sunshine (S) Another problem using random samples may be described. At every climatological station, cloudiness is observed. At a few of them, bright sunshine is recorded continuously. It goes without saying that the average monthly duration of sunshine is most interesting information for many purposes. FIG. 11. The relation between cloudiness and duration of sunshine. E abcdef W = path of the sun. The parts of the path ab } cd, ef, are covered by clouds. The question of how to calculate sunshine duration from aver- age cloudiness is approximately solved by means of random sam- ples. We assume that the path of the sun (Fig. \\.Eabcdef W} for ajjiyen day of the year is obscured by clouds for the portions a6> cd> ef. If these parts are known from the records of a sunshine recorder, a simple calculation gives the percentage, C, of the total 56 METHODS IN CLIMATOLOGY length of the path of the sun which is covered by clouds. It is clear that the rest of the path S = 100 - C. Now, if the assumption is made that the conditions along the path of the sun are a random sample, valid for the entire visible sky, the problem is solved in principle. In reality, this hypothesis cannot be made for an individual day, only for an average day within a period of at least one month. Even then, the assumption remains a rough approximation, as will be grasped from the ex- amples in Table 12. They show that the annual and monthly values of S and C yield sums which indeed generally differ less than 10% from 100. This agreement is not bad. The simple formula, S + C = 100%, has been improved by theoretical considerations. The resulting formula is not tested TABLE 12. THE EQUATION S + C = 100% COMPARED WITH OBSERVATION RESULTS IN CALIFORNIA AND IN THE MEDITERRANEAN REGION (S in % of possible duration; C in % of the visible sky) Jan. July Year % % % San Francisco, Calif. S: 53 69 65 37.8N, 122.7W C: * 54 37 42 S + C: 107 106 107 San Diego, Calif. S: 67 68 68 32.7N, 117.2W C: 40 34 36 S + C: 107 102 104 Sacramento, Calif. 38.6N, 121.4W S: C: 46 59 96 08 74 31 S + C: 105 104 105 Messina, Italy S: 37 75 52 38.2N, 15.6E C: 61 28 46 5 + C: 98 103 98 Athens, Greece S: 49 81 60 38.0N, 23.7E C: 55 11 40 5 + C: 104 92 100 Helwan, Egypt S: 70 90 82 29.9N, 31.1E C: 41 06 23 S+C: 111 96 105 SPECIAL STATISTICAL CHARACTERISTICS 57 for different climates, and its evaluation is relatively complicated, so that, generally, the climatographer would prefer not to use it. 22 Finally, in connection with this problem, another definition may be mentioned : B = 100 - C where C is again the cloudiness in per cent of the visible sky ; B can be called the brightness of the sky, and is used sometimes in meteor- ological and climatological papers. 11 See V. Conrad, " Die klimatologischen Elemente und ihre Abhangigkeit von terrestrischen Einfltissen," Handbuch der Klimatologie, vol. I (Berlin, 1936), p. 450, and A. Wagner, "Beziehungen zwischen Sonnenschein und BewOlkung in Wien," Meteoro- logische Zeitschrift, 1937, pp. 161-167. CHAPTER IV SOME PROBLEMS OF CURVE FITTING AND SMOOTHING OF NUMERICAL SERIES IV. 1. THE STRAIGHT LINE IN CLIMATOLOGY, one is often confronted with the. problem of representing an element as a function of an independent variable, such as altitude, latitude, distance, etc. Each value of the variable element is related to a certain value of the inde- pendent variant. The first step is to arrange the values of the element according to increasing values of the independent variable. If there is a greater number of values, it may be advantageous to divide the data into groups, and to average either variant within each interval. The variables are plotted on the X- and F-axis, respectively, and a line is drawn freehand through the resulting points. This procedure may suffice for survey of the correlation of both variants. In any case, the type of* curve can be recognized; in climatology, one may often assume it to be a straight line. TABLE 13. THE RELATIONSHIP BETWEEN THE AMPLITUDE OF THE ANNUAL VARIATION OF AIR PRESSURE (ai, in mm mercury) AND THE ALTITUDE ABOVE SEA LEVEL (h in hectometers; 1 hectometer = 100 m) (1) (2) (3) (4) (5) (6) h a\ a\ a\ (calculated by the semi-average (mm Hg method Obs. calc. (calc. with No. (hm) observed) a\ =0.29 +0.16/0 (2-3) 01 =0.36 +0.16/0 Obs. -calc. 1 0.5 0.28 0.37 -.09 0.44 -.16 2 2.7 1.08 0.72 + .36 0.79 + .29 3 4.6 1.04 1.02 + .02 1.09 -.05 4 6.1 1.24 1.26 -.02 1.33 -.09 5 7.8 1.38 1.54 -.16 1.61 -.23 6 9.0 1.72 1.73 -.01 1.80 -.08 7 16.4 3.04 2.91 + .13 2.98 + .06 8 23.4 4.26 4.03 + .23 4.10 + .16 9 30.5 5.34 5.17 + .17 5.24 + .10 2(+) +91 + 61 s(-) -28 -61 +63 58 SOME PROBLEMS OF CURVE FITTING 59 An example illustrates this problem : Records of the whole-year amplitudes of air pressure at 3 1 stations are correlated with the alti- tude of the respective places. The amplitudes are arranged in 9 groups according to increasing heights. The averages (hectom- eters) of the intervals are indicated in column 1 of Table 13. The averages of the respective amplitudes are contained in column 2. 30 28 28 24 22 20 18 I- ie to 8 8 4 t z 1.0 2.0 3.0 4.0 AMPLITUDE O.I mmHg 8.0 FIG. 12. Annual range of air pressure (abscissa) and altitude (ordinate). (After V. Conrad) Starting from these two series of numbers, one derives the points marked on the graph (Fig. 12). A straight line fits the ob- servations best. The analytical equation of a straight line has the form : ai = A + Bh where a\ means the amplitude of the whole annual variation, h the altitude, and A, B, constants which have to be evaluated. For this reason 9 equations are available from Table 13. 60 METHODS IN CLIMATOLOGY No. 1 0.28 = A + 0.5 B No. 2 1.08 = A + 2.7B No. 9 5.34 = A + 30.5 B Here is just the case to use the method of least squares. As was said above, however, as a rule it does not pay to apply this rather wearisome method to climatological problems. Usu- ally, the accuracy of the observations is not sufficient to use least squares. 1 Therefore a method of quick approximation is given 1 In the case of a linear relation, the method of least squares yields relatively simple equations for the constants A and B of the equation a = A + Bh, the computation of which is not too laborious. Sfr-S(qft) - 2a-2fr* _ Sft-2a - nZ(ah) These formulas, applied to the example of Table 13, yield: (1) for the calculation of the numerator of A : Xh = 101.0; 2(aA) = 354.158; 2h-X(ah) = 35769.958; Sa = 19.38; S 2 = 1954.52 2a-2^ = 37878.5976 (2) for the calculation of the numerator of B: t Sfc-Sa = 1957.38; w-Sfa/i) * 3187.422 (3) for the denominator of A and B : (Sfc) 2 = 10201.0; n-2fc' = 17590.68 and finally: A = + 0.2853 B = + 0.1664 so that the equation, got by the method of least squares, reads: ai = 0.285 + 0.166& (I) The semi-average method, given in the text, resulted in: 0! 0.29 + 0.16A (II) The values of A are nearly identical, the values of B differ for about 4%. For further correction, the straight line defined by equation (II) is shifted parallel to itself increasing A from 0.29 to 0.36 and the equation reads ai 0.36 + 0.16A (III) This step is advantageous in so far as it makes the sum of the positive deviations equal to that of the negative ones by a minimized arithmetic. In the present example it would have been more nearly correct to leave the constant A unchanged and to diminish the angle between the straight line and the abscissa as it can be seen from the exact equation (I). On the whole, the agreement between the approximations given by the semi- average method and similar methods, on the one hand, and the exact result of the method of least squares, on the other hand, is generally sufficient for climatological purposes. The reader who takes pains with making a diagram on graph paper to a scale even twice that of Figure 12 will become aware of how close the three lines run to each other. SOME PROBLEMS OF CURVE FITTING 61 here instead, the semi-average method. The first half and the second half of the values of both variables are averaged, so that two pairs of coordinates result, which have to be introduced into the general equation of a straight line. In the special case of Table 13, the number of items is relatively small and odd. In order to avoid asymmetry and unequal weight, item no. 5 can be included both in the first and in the second half. The first half then contains items nos, 1 to 5 ; and the second half, nos. 5 to 9. The two equations read : 1.004 = A + 4.345 3.148 =4 + 17.42 B Therefore A = 0.29 and B = 0.16 and the straight line is given byai = 0.29 + 0.16 A. The nine values of h (Table 13, nos. 1 to 9) introduced into the equation yield the calculated values of #1 in column 3. Column 4 contains the differences: Observed values (obs.) minus calculated values (calc.). It is seen that (at the bottom of the table) the positive differences exceed the negative ones by 0.63. This means, that the observed a\ are, on the average, 0.07 mm Hg greater than the calculated. Therefore, we have to shift the calculated straight line parallel to itself to the right side by 0.07 mm: in other words, increase the constant term 0.29 by 0.07. The definitive equation now reads: ai = 0.36 + 0.16 h The new values computed with this equation appear in column 5, and the differences "obs. calc." in column 6. Incidentally, the sum of the differences (with regard to the signs) is zero, which represents the ideal. The process of shifting has to be continued until the sum becomes a minimum. 2 In many cases it is not even necessary to use the semi-average method. When the number of observations varies greatly from one class interval to the next, it is sometimes more advantageous to take the averages of only two intervals with the greatest number of 2 The example is taken from V. Conrad, "The Influence of Altitude on the Yearly Course of Air Pressure," Bulletin of the American Meteorological Society, vol. 20 (1939), p. 207. 62 METHODS IN CLIMATOLOGY observations, in such a way that the midpoints of the intervals may be as far apart as possible. 3 IV. 2. DECAY-CURVES In climatology, as in some other fields, the observation of the decay of any quantity intensity, ratio, etc., with time, distance, height, or depth, has to be represented by an analytical equation. Example: In a problem dealing with cloudiness 4 the decay of a ratio y is to be described which is infinite at the place of observa- tion and decays to zero with increasing distance. These boundary conditions lead to the equation y = Ax~ B In terms of logarithms: log y = log A B log x Thus, the exponential equation is reduced to a linear equation regarding the two unknown constants log A and B. Plotting on logarithmic paper yields the values A and B in more or less rough 2.00 1.80 1,60 140 1.20 30 40 60 80 100 DISTANCE km.x 120 FIG. 13. Observations represented by the formula y (After V. Conrad) Ax -B * For an interesting case of curve fitting, see Hans Neuberger, "Studies in Atmos- pheric Turbidity," Pennsylvania State College Studies, no. 9 (1940), p. 22. There, a dot chart is presented which seemingly offers a complex problem. In reality, it is easily solved by the assumption of two straight lines instead of one curve of higher order. 'Taken from V. Conrad, "Zum Studium der Bewolkung," Met. Z.S., 1927, p. 87. SOME PROBLEMS OF CURVE FITTING 63 approximation. On the other hand, the problem is reduced to that of the straight line. This has been discussed above. Therefore the numerical example in Table 14 and in Figure 13 does not need further explanation. TABLE 14. THE REPRESENTATION OF OBSERVATIONS BY AN EXPONENTIAL CURVE y = Ax~ B (1) (2) (3) (4) Distance * (km) Ratio y obs. Ratio y calc. Obs. - calc. 21 2.09 2.10 -.01 48 1.65 1.69 -.05 80 1.60 1.48 + .12 123 1.27 1.33 -.06 2(+) + .12 s(-) -.12 The solution of the logarithmic equation yields : A = 4.618 B = 0.2590 and the equation reads : y = 4.61 8 x-o.2590 IV. 3. THE EQUATION y = B-e Ax If the dependent variable, y, has a finite value at the origin and increases with increasing x to infinite values, the equation y = B-e Ax may often represent the observations in a suitable way. Plotting on semilogarithmic paper yields the values of A and B. In logarithmic terms the equation reads: log y = log B + A x log e where "log" means common (Briggsian) logarithms, e = 2.718, log e = 0.4343. The logarithmic equation has a linear form so that A and B can be easily determined. No general rule can be given for the choice of the right form of equation. The graphical representation remains the best guide. A collection of different forms of equations for curve fitting can be 64 METHODS IN CLIMATOLOGY found in various books dealing with numerical calculations and mathematical statistics. 6 IV. 4. THE QUADRATIC EQUATION Figure 14 represents an example of a group of curves which is also important in the discussion of climatic problems. The ob- servations averaged for 5 class intervals show the behavior of HM FlG. 14. Observations represented by a quadratic equation (x = A + By + Cy*) (After V. Conrad) E.g., in H. L. Rietz, Handbook of Mathematical Statistics (New York, 1924), where also the transformation of the different equations to linear forms is given. Different forms of the equation for the hyperbola (e.g. xy = const, or y 2 = x(m + nx) are often very suitable for the representation of the correlation between climatological elements and factors. Examples are given in V. Conrad, " Messung und Berechnung der AbkiihlungsgrGsse" (Measuring and Calculating the Cooling Power; with an English summary), Gerlands Beitrage zur Geophysik, vol. XXI, 1929, pp. 183-189. Another example of curve fitting by means of hyperbola equations can be found in V. Conrad, and O. Kubitschek, " Die Veranderlichkeit und Machtigkeit der Schneedecke SOME PROBLEMS OF CURVE FITTING 65 average winter temperature at different average heights above the bottom of an Alpine valley. 6 The well-known phenomenon of the inversion is illustrated by means of the numbers in Table 15 and by Figure 14. TABLE 15. OBSERVATIONS REPRESENTED BY THE QUADRATIC EQUATION: x = A + By + Cy (VARIATION OF TEMPERATURE WITH HEIGHT IN AN ALPINE VALLEY IN WINTER) (h = hectometers, / in C) (1) (2) (3) (4) h fC /c No. (hm) (observed) (calculated with /=-8.1-fl.06A-0.05A) Obs. -calc. 1 4.9 -4.0 -4.1 +0.1 2 6.7 -3.1 -3.2 +0.1 3 9.0 -2.5 -2.6 +0.1 4 11.3 -2.9 -2.5 -0.4 5 19.0 -5.8 -5.9 +0.1 2(+) +0.4 2(~) -0.4 In cases like this, it is not easy to determine the exact nature of the curved line; so the use of a quadratic equation is recommended. The form is: x = A + By + Cy 2 where y is the independent variable. According to Table 15, five pairs of values can be used for com- puting the coefficients A, B, C. Even in this problem, the use of the method of least squares can be avoided if the observed values lie in a smoothed curve, as is seen in the actual example. Then a good approximation can often be reached by choosing only three equations for computing the three constants. Naturally, points on the curve which charac- in verschiedenen Seehohen" (Variability and Depth of Snow on the Ground in Different Altitudes, with an English summary), Gerlands Beitrdge zur Geophysik, vol. LI, 1937, pp. 100-128. In the East Alps, the variation of the depth of snow on the ground with altitude can be well represented by hyperbolas from November through March. In April, and this is of great methodological interest, the hyperbola must be replaced by a parabola, which considers the melting of the snow cover in the lower levels in spring, on the one hand, and a further increase of the depths of snow in the high levels, on the other hand. The hyperbola reproduces the relations between depth of snow and alti- tude as far as in every level an increase of the depths takes place. 8 Example from V. Conrad, Klimatographie von Karnten (Wien, 1913). 66 METHODS IN CLIMATOLOGY terize the trend as well as possible should be selected from the graph. In the present example, the following equations were chosen: -4.0 = A + 4.9 B + 24 C -2.5 = A + 9.0 B + SIC -5.8 = A + 19.0 B + 361 C These are the pairs of values nos. 1,3,5. The equations are easily solved. It is necessary to change the term A by only 0.1 C to equalize the sum of positive and negative differences "obs. calc." It follows the equation : / = - 8.0 + 1.05 h - 0.05 A 2 (C) where t means the temperature, h the height in hectometers. The constants are : A = - 8.0; B = 1.05; C = - 0.05 The differences "obs. calc." show clearly that, in general, in a climatological problem greater accuracy is unnecessary ; another choice of intervals, another choice of stations, etc., would cause slight changes of the vertical temperature distribution which are of the same order of magnitude as the differences "obs. calc." Therefore the application of more exact methods would be a waste of time. The advantages of representing variations of elements by analytical equations are obvious. The equations permit an objec- tive interpolation, based on the totality of observations. IV. 4. a. Level of the Turning Point of the Curve As a supplement to the last problems, two applications of the representation of observations by means of analytical equations are given. From the observations (see Fig. 14), it is clear that the temperature increases in the lower layers with height (inversion), and above a certain level the normal decrease of temperature occurs. Question : At what altitude is the turning point of the tempera- ture-height curve located? SOME PROBLEMS OF CURVE FITTING 67 At the extremes, the differential quotient of a function becomes zero. ^ = 1.05 - 2 X 0.05 h = 1.05 - 0.1 h dh 3/J = 1.05 - 0.1 A = dfl / t~max and hence (/*)<-ma* = 10.5 hectometers, or = 3445 ft. This level is the average top-level of the inversion in winter in the mountain valley in question. IV. 4. b. The Standard Distribution As was said before, the analytical equation permits full inter- polation of the element for every value of the independent variable. In the present example, the average winter temperature can be given for every desired altitude which lies between the extreme levels of the places of which temperatures are used in the com- putation. For some purposes, it is very important to know the average temperatures at the consecutive levels, for instance, 10-meter or 100-meter levels, above the bottom of the valley. A clipping from such a table calculated from the equation t = - 8.0 + 1.05 A -0.05/t 2 is given in Table 16. It indicates the average temperatures every TABLE 16. CLIPPING FROM A TABLE OF AVERAGE DISTRIBUTION OF TEMPERATURE ABOVE THE BOTTOM OF A MOUNTAIN VALLEY IN JANUARY Altitude 00 10 20 30 40 etc. meters Centigrade degrees 400 -4 .6 -4, 5 -4. 5 -4 .4 -4.4 500 -4 .0 -3.9 -3. 9 -3 .8 3 ft "^ O.O 600 -3 .5 -3. 5 -3. 4 -3 .4 -3.3 etc. 10 m. The content of such a table represents the standard distribu- tion of temperature with height, in the region in question. This representation of the distribution of temperature with height is naturally only one example out of many. The method can and should be applied to any climatological element which is 68 METHODS IN CLIMATOLOGY correlated with a climatic factor, or with another climatic element. (See XIV, 4 and 5.) IV. 5. SMOOTHING OF NUMERICAL SERIES This procedure is usual in climatological practice, if the aver- aged variation of an element for instance, an average annual course presents an irregular aspect. One assumes that the ir- regularities are of incidental nature and would diminish in proportion to the length of the available period. Different methods are applicable. Some, of greater importance for the climatologist, are mentioned below, in each of which a series of the following ten numbers will be used for a numerical example: 12, 31, 9, 4, 11, 10, 15, 7, 11, 13. 1) Overlapping sums: (a) of three consecutive elements according to the formula : di = (#;_! + 0; + 0;+i) 3 where means any element in the series, the subscript (i 1) stands for the preceding, (i + 1) for the following, element. di is the smoothed value replacing thfe^original a*. Example: 01, 02, 03, 04, 05, 06, 07, 08, 09, 010 12, 31, 9, 4, 11, 10, 15, 7, 11, 13 The series may be considered as cyclic (for instance, an annual course). Then, the smoothing procedure begins with the last item and ends with the first, and the smoothed series reads: 01 = (010 + 0i + 02): 3 = (13 + 12 + 31): 3 = 19 2 = (01 + 02 + 03): 3 = (12 + 31 + 9): 3 - 17 010 = (09 + 010 + 01): 3 = (11 + 13 + 12): 3 = 12 so, finally, we have the sequence of smoothed values : 19, 17, 15, 8, 8, 12, 11, 11, 10, 12 (b) The number of overlapping consecutive items can be in- creased arbitrarily, considering the total number of items of the series. For simple reasons, odd numbers should be preferred. As a second example, the formula for 5 consecutive items is given: 0; = (0i-2 + fli-i + + 0;+i + 0*4-2) SOME PROBLEMS OF CURVE FITTING 69 The item di of the series (la) is therefore: a\ = (ag + aio + #1 + #2 + #3) : 5 = (11 + 13 + 12 + 31 +9): 5 = 15 a 2 = (#10 + fli + #2 + #3 + #4) : 5 = (13 + 12 + 31 + 9 + 4) : 5 = 14 etc. 2) Weighted overlapping sums: (a) <z* = (a;_i + 2#i + a t -+i) : 4 Numerically : #1 = (^10 + 2&i + a 2 ) : 4 = (13 +2 X 12 + 31): 4 = 17 etc. (b) a* = (a t -_ 4 + 2a l - 3 + 3a v _ 2 + 4ai_i + Sa t - +1 . 25 ai+4 + 2a i4 . 3 + 3a i+2 + 4a i+ i) j ' Numerically : di = (a 7 + 2a 8 + 3a 9 + 4ai + ^ a i + 1 . 25 a 5 + 2a 4 + 3a 3 + 4a 2 ) j ' fii = (15 + 14 + 33 + 52 + 60 + 11 + 8 + 27 + 124) = 14 etc. (c) A more individual weighting is obtained by using the bino- mial coefficients as weights for the consecutive items. Formula 2a represents the simplest case, the binomial coefficients of a squared binom : i.e., 1, 2, 1 The binomial coefficients for the 4th power are: 1, 4, 6, 4, 1; and those for the 6th power: 1, 6, 15, 20, 15, 6, 1. Therefore, the smoothing formula, corresponding to the 4th power, is: di == (ai_ 2 + 4a i _ 1 + 6a.- + 4a i+1 + a i+2 ) : 16 The first term of our numerical series when smoothed is: di = (ag + 4aio + 6#i + 4a 2 + #3) : 16 = (11 + 52 + 72 + 124 + 9): 16 = 17 Numerical series should be smoothed only if absolutely neces- sary. The smoothing procedure can efface essential characteristics of the series and lead to false conclusions. CHAPTER V HARMONIC ANALYSIS V. 1. THE ANALYSIS THE HARMONIC ANALYSIS is one of the methods for describing periodic phenomena. The latter are indicated by a varia- tion in which the dependent variable shows a repetition at equal intervals of the other variant. A great number of climato- logical elements reach one extreme value at about low sun and the opposite extreme at about high sun. The periodical variation of the altitude of the sun is the physical cause of the periodical annual variation of these elements. Their daily course frequently offers a complex problem. The physical conditions during the day are different from those during the night. A sound physical stand- point should never be neglected in investigations of periodical phenomena. The harmonic analysis makes use^D/ Fourier's series. We start from the following formula, which is rather common in meteor- ological and climatological investigations: y = a<> + #1 sin (x + AI) + a* sin (2x + A%) + + a* sin (ix + Ai) + where a is the arithmetic mean of the observations and #1, a 2 , a* are amplitudes (half ranges) of the superimposed waves. Ai, At, -Ai are called phase angles, determining the times at which the extremes occur. The meaning of the equation mentioned is easy to understand by means of the graph of Figure 15. The symbol x represents the time angle. It is computed from 360 if P denotes the length of the period investigated. For example, if P = 12 months, then * = TT = 30 ' 2x = 60 ' etc ' 70 HARMONIC ANALYSIS 71 30 60* 90* 120 150 180 2JO 240 270* 300* 330 360* FIG. 15. Example of superposition of two waves: I = aisin (x + Ai)\ AI = II = a 2 sin (2x + AJ)\ A 2 = 90 III = sum of the waves I and II If P = 24 hours 360 1 co x = -=-r- = 15 etc. 24 Further, it is easy to see that y = a + a\ sin x cos A i + a 2 sin 2x cos -42 + + &i cos x sin .4 1 + a 2 cos 2# sin ^4 2 + We substitute : = a\ sn <r, = ai cos A\\ = ^2 sn =^2 cos Then, y = a + pi cos # + 2 cos 2x + + gi sin # + g 2 sin 2x + w , tf i, #i w n _i may be the deviations from the arith- metical mean and are inserted into the equation in place of y. Then we get n equations from which the constants pi, qi, pi, 32 can be calculated. If the number of observations is greater than that of the con- stants (the common case), the method of least squares is to be applied. This method yields the following equations for the 72 METHODS IN CLIMATOLOGY constants : pi = 2/n[_u cos + #1 cos x + Uz COS 2tf + ' ' ' ^n-l COS (ft w sin + u\ sin x -f* ^2 sin 2# -}- u n \ sin (w "w cos + HI cos 2x -|- ^2 cos 4# -[- w n _i cos 2 (ft sin + u\ sin 2x + 2*2 sin 4# + w n -i sin 2 (ft o COS + Wi COS kx + 2/2 cos 2&# + u n -i cos (ft t, sin + ^i sin kx + w 2 sin 2^ + ^n-i sin (ft - According to the definitions of p k and g>, ^ fc j A $* k Obviously, also, the following solutions are valid : L. W. Pollak J introduced an excellent procedure for evaluating the equations mentioned, which is so clear and simple that even those without special mathematical training are able to make the necessary calculations. In the following, therefore, Pollak's system is illustrated. For lack of space, only two special problems, frequently used in climatology, can be discussed : 1) 12 equidistant observations (average values) 2) 24 equidistant observations Case 1 usually represents the annual course of an element; case 2, the diurnal course. There is, however, no difficulty in calculating the constants of a diurnal course with 12 two-hourly 1 L. W. Pollak, "Handweiser zur harmonischen Analyse/' Prager Geophysikalische Studien, vol. II (Prague, 1928); L. W. Pollak, Rechentafeln zur harmonischen Analyse, (Leipzig, 1926). I learn from a letter that Dr. Pollak is extending the Rechentafeln up to 100 ordinates. The first edition reached 40 ordinates. HARMONIC ANALYSIS 73 values and in evaluating an annual variation by means of 24 half- monthly means, if greater accuracy is required. Generally, the monthly means are related as was said above to the middle day of the month: e.g. the 14th, 15th, or 16th day of TABLE 17. INSTRUCTIONS FOR CALCULATING THE CONSTANTS FOR 12 EQUIDISTANT VALUES YIELDING THE FIRST, SECOND, AND THIRD TERM OF FOURIER'S SERIES (After L. W. Pollak) i Pi 01 P* 0J P* 01 1 1 1 1 0.866 0.5 0.5 0.866 1 2 0.5 0.866 -0.5 0.866 -1 3 1 -1 -1 4 -0.5 0.866 -0.5 -0.866 1 5 -0.866 0.5 0.5 -0.866 1 6 t 1 -1 7 -0.866 -0.5 0.5 0.866 -1 8 -0.5 -0.866 -0.5 0.866 1 9 -1 -1 1 10 0.5 -0.866 -0.5 -0.866 -1 11 0.866 -0.5 0.5 -0.866 -1 TABLE 18. THE SAME AS IN TABLE 17, BUT FOR 24 EQUIDISTANT VALUES (After L. W. Pollak) f Pi Qi 1 0* P* 1 1 1 1 0.966 0.259 0.866 0.5 0.707 0.707 2 0.866 0.5 0.5 0.866 1 3 0.707 0.707 1 -0.707 0.707 4 0.5 0.866 -0.5 0.866 -1 5 0.259 0.966 -0.866 0.5 -0.707 -0.707 6 1 i -1 7 -0.259 0.966 -0.866 -0.5 0.707 -0.707 8 -0.5 0.866 -0.5 -0.866 1 9 -0.707 0.707 -1 0.707 0.707 10 -0.866 0.5 0.5 -0.866 1 11 -0.966 0.259 0.866 -0.5 -0.707 0.707 12 -1 1 -1 13 -0.966 -0.259 0.866 0.5 -0.707 -0.707 14 -0.866 -0.5 0.5 0.866 -1 15 -0.707 -0.707 1 0.707 -0.707 16 -0.5 -0.866 -0.5 0.866 1 17 -0.259 -0.966 -0.866 0.5 0.707 0.707 18 < -1 1 19 0.259 -0.966 -0.866 -0.5 -0.707 0.707 20 0.5 -0.866 -0.5 -0.866 -1 21 0.707 -0.707 1 -0.707 -0.707 22 0.866 -0.5 0.5 -0.866 -1 23 0.966 -0.259 0.866 -0.5 0.707 -0.707 74 METHODS IN CLIMATOLOGY the month, depending on the length of the respective months. Hourly means are related to the 30th minute of the hour. Tables 17 and 18 contain instructions for calculating the con- stants Ply ffli 2, 22, p3, ?3, and from these, #1, #2, #3 and Ai 9 A% 9 Aa The first column of each table contains the number of the item. For example: the mean of May has the number i = 4, because January has the number zero. The value of 5 A.M. has number i = 5 ; if the series begins with midnight = hour. In columns p\, qi. p^ 2 the factors by which the respective items (see Column i} have to be multiplied, are indicated. The columns pi, qi, etc., are totaled (naturally, with regard to sign). 2 The sums multiplied by 2/n (that is, 1/6 for 12 values, and 1/12 for 24 values) yield the constants pi, qi, etc. The multiplica- tions are done with the aid of a slide rule, a multiplication table, or a calculating machine. The best expedient is Pollak's Rechen- tafeln, which are indispensable if such calculations are made on greater scale, or if other lengths of period are considered. In these calculations, however, one shoufcl shun a purely arithmetical accuracy not guaranteed by the exactness of the observations. Generally, results to three places of decimals are ample. If the constants p and q are calculated, the amplitude a and the phase angle A, are easily computed with the formulae given above. Although the problem is exceedingly simple, blunders some- times happen in determining the true phase angle. The diagram (Fig. 16), helps to avoid such mistakes. Possible combinations of the signs of p and q for which the phase angle is to be determined, are: a) +P + q; b) + p-q\ c) - p - 2; d) - P + q 1 The deviations u<>, ui, ,t/ n -i are multiplied with the factors of each of the columns p\, q\\ p*, g 2 ; . These series of products are entered into a new blank, with the same headings pi, q\\ p^ q^\ -. The column p\ of the new blank contains, thus, the following products (in the case of Table 17): 1 X w 0.866 X ui 0.866 X ttn-i etc. These columns p\ t q\\ P*,qi\ are then totaled. HARMONIC ANALYSIS 90 75 I80l 270 FIG. 16. Diagram for determining the true phase angle From these combinations, we calculate directly the following angles : a) A. b) c) d) Then the true phase angle A is : a) A = -<4++ A in the first quadrant b) A = 180 A+- Am the second quadrant c) A = 180 + A __ A in the third quadrant d) A = 360 A-+ A in the fourth quadrant The following example, 3 Table 19 shows each step of the calcu- lation, so that a few explanatory remarks are sufficient. Taken from V. Conrad, "Anomalien und Isanomalen der Sonnenscheindauer in den oesterreichischen Alpen," Beihefte Jahrbuch d. Zentralanstalt f. Meteorologie (Wien, 1938); V. Conrad, "Die Komponenten der Jahresschwankung der Sonnenscheindauer/' Helvetia Physica Acta, vol XII (1939), p. 38. 76 METHODS IN CLIMATOLOGY TABLE 19. HARMONIC ANALYSIS OF THE ANNUAL COURSE OF THE RELATIVE SUNSHINE DURATION (S in %) AT AN AVERAGE LEVEL OF 1000 FEET IN A MOUNTAINOUS COUNTRY (di = deviations) (1) (2) (3) (4) (5) (6) (7) (8) Month 5(%) i di Pi Qi P* q* I 25 -19 - 19 -19 II 39 1 - 5 - 4.330 - 2.5 - 2.5 - 4.330 III 47 2 + 3 + 1.5 + 2.598 - 1.5 + 2.598 IV 45 3 + 1 + 1 - 1 V 53 4 + 9 - 4.5 + 7.794 - 4J5 - 7.794 VI 58 5 + 14 - 12.124 + 7 + 7 -12.124 VII 62 6 + 18 - 18 + 18 VIII 58 7 + 14 - 12.124 - 7 + 7 + 12.124 IX 54 8 + 10 - 5 - 8.660 - 5 + 8.660 X 41 9 - 3 + 3 + 3 XI 29 10 -15 - 7.5 + 12.990 + 7.5 + 12.990 XII 19 11 -25 - 21.650 + 12.5 -12.5 + 21.650 mean (flo) 44 -102.728 + 28.722 - 3.5 +33.774 Pi - - 17.121 kPi = 1.2335 pi = - 0.583 lg/>2 = 9.7657 tfi = + 4.787 lgi = 0.6801 q z = + 5.629 lg$2 = 0.7504 (Ai) = 7422' Ig tan A i = 0.5534 (,4 2 ) = 555' Ig tan A* = 9.0153 a\ = 17.8% kPi = 1.2335 . a 2 = 5.7% lg/>2 = 9.7657 A, = 28538' Ig sin A i = 9.9836 w4 2 = 3545' Ig sin A 2 = 9.0132 Ig ai = 1.2499 Ig a 2 = 0.7525 S 44 + 17.8 sin (x + 286) + 5.7 sin (2* + 354) Column 2 contains the original numbers, the periodical trend of which is to be analyzed. Columns 5 to 8 contain the products of the consecutive deviations (see column 4), and of the values of the trigonometrical functions of the respective angles as indicated in Table 17. V. 2. EVALUATION OF THE EQUATION Table 20 gives a pattern of the evaluation of Fourier's series: 5 = 44 + 17.8 sin (x + 286) + 5.7 sin (2* + 354) This evaluation is to be recommended, since it is the only check on the correctness of the analysis. The graph, Figure 17, furnishes a good illustration of the agreement of observation and calculation. The example shows, too, that the combination of only two waves is sufficient to give a good approximation with the rather complex HARMONIC ANALYSIS TABLE 20. EVALUATION OF EQUATION 5 IN TABLE 19 77 (1) (2) (3) (4) (5) (6) (7) (8) (9) Month * (x + A0 2x -{-At 18 sin (x -f -4i) 6 sin (2* -|- A ) C. O. o-c Jan. -7422' - 555' -17.1 -0.6 -18 -19 -1 Feb. 30 -4422' + 5405' -12.4 +4.6 - 5 +3 March 60 -1422' +6555' - 4.4 +5.2 + i + 3 +2 April 90 + 1538' + 555' + 4.8 +0.6 + 5 + 1 -4 May 120 +4538' -5405' + 12.7 -4.6 + 8 + 9 + 1 June 150 + 7538' -6555' + 17.2 -5.2 + 12 + 14 +2 July 180 + 7422' - 555' + 17.1 -0.6 + 17 + 18 + 1 Aug. 210 +4422' + 5405' + 12.4 +4.6 + 17 + 14 -3 Sept. 240 + 1422' +6555' + 4.4 + 5.2 + 10 + 10 Oct. 270 -1538' + 555 ; - 4.8 +0.6 - 4 - 3 + 1 Nov. 300 -4538' -5405' -12.7 -4.6 -17 -15 +2 Dec. 330 -7538' -6555' -17.2 -5.2 -22 -25 -3 2(+) - 12 s(-) - 11 C. = Calculated deviations. O. = Observed deviations. I II III IV VI VII VIII IX XI XIJ I tio o -10 -20 + 10 -10 FIG. 17. At the top: the annual course of the relative duration of sun- shine in a given average level of a mountainous region. Full line: calculated. Dots: observed. At the bottom: the two constituents of the calculated curve. 78 METHODS IN CLIMATOLOGY observed curve. As far as the approximation is concerned, atten- tion should be directed to the convergence of the resulting series. The ratio - = 1/3, and - - 1/9 (a, not included in Table 19), di di so that here a good convergence exists. Thus, the series can be cut off at the second term, in the example of Table 19. 4 V. 3. RELATIVE AMPLITUDES The amplitudes #1, a 2 , a 3 , are given in the units of the devia- tions. The meaning of the amplitudes is dependent upon the particular arithmetical mean. For purposes of comparison, the amplitudes of the different terms have to be freed from the in- fluence of the unit and that of the arithmetical mean: to do this, the amplitudes are divided by the arithmetical mean. d\ d% In the example (Table 19) : d R>l = 0.40 a*, 2 = 0.13 t Often it is advantageous to give the relative amplitudes in per cent of the arithmetic mean. Then a Rt i = 40%, a*, 2 = 13% V. 5. TIMES OF THE EXTREMES The times of the extremes of the single constituent waves are easily calculated from the equation : kx + A k = 90 or 450 respectively, 4 D. Brunt (Meteorological Magazine, 1937, p. 268), gives a criterion as to whether or not it pays to calculate further terms of the Fourier's series. Brunt's formula reads: where <r 2 means the variance of the value (di, column 4 in Table 19) subject to the analysis In the example ff . = 5*'=179 n i(ai* + a, 8 ) 175 Therefore a' 2 = 4 This means that cr /2 is only 2% of a 2 . Hence the criterion indicates that further terms of the series can be neglected. HARMONIC ANALYSIS 79 where k equals the number of the term. Example from Table 19: L ~~ t /Time of maximum = x + A\ = x + 286 = 450 or k = 1 J *ma* = 164 * m in = 164 + 180 = 344 Since 360 degrees correspond to an average year of 365.25 days, the angle-degrees have to be multiplied by 1.0145, if the conversion into days is required. The date is obtained by adding 16 to the product. This is clear, because the mean values of the elements are related to the midpoint of the month (generally January). Example: If x = 164, 1.0145 X 164 = 166 days have been passed between the middle of January and the time of the maxi- mum. 6 Therefore the maximum of the whole-year-wave occurs on the 166 + 16 = 182nd day of the year: that is, July I. 6 It goes without saying that the starting date can be chosen arbitrarily. The hydrologist, for instance, begins his year on July first, October first, November first, or any other date repre- senting the beginning of the chief rainy season. The problem of calculating the time of the extremes directly from Fourier's series with two terms was simply solved nearly a century ago. The solution gives a good approximation. 7 Gen- erally, it is sufficient to take the times of the extremes from a good graph. In our example, the curve representing the annual course of the relative duration of sunshine (Fig. 17) indicates, approxi- mately, August 1 for the maximum and December 16 for the minimum. Harmonic analysis permits the interpolation of missing or dubious values of a periodical series and is an excellent means of investigation in many problems of periodicity. The example of Table 19 is taken, for instance, from an investigation in the varia- tion of the annual course of the duration of sunshine with altitude. By means of the harmonic analysis, it was possible to puzzle out the complex problem and to show among other results that the amplitude of the second harmonic term is invariant with height. 8 In order to illustrate the high research value of the harmonic an- 8 It is sufficient to increase the number of degrees by li%. * See Appendix IV. 7 Hugo Meyer, Anleitung zur Bearbeitung meteor ologischer Beobachtungen (Berlin, 1891), p. 38 ff. 8 This result is important in so far as it indicates that, against every expectation, the half-year wave is the main wave upon which the whole-year wave of very variable am- plitude and phase angle is superposed. 80 METHODS IN CLIMATOLOGY alysis, attention is drawn to Hann's excellent studies of the daily course of air pressure, as also to the numerous investigations which deal with the idea of Arthur Schuster's "expectancy/* and to the methods of L. W. Pollak and Sir Gilbert Walker. Even if the climatologist is not compelled de facto to use the harmonic analysis for his own work and investigations, he must understand this method, so often applied in scientific literature. PART II REPRESENTATION OF CHARACTERISTIC FEATURES OF DIFFERENT ELEMENTS IT is NEITHER desirable nor possible to give detailed instructions as to how to handle each element and its characteristics. The first part of this book has offered the mathematical means of descriptive statistical characteristics for every series of numbers, independent of the element. The only restriction already men- tioned was that here scalar quantities, not vectorial, were considered. Many of the common characteristics have been discussed pre- viously. A partial summary of the most usual characteristics is given in the forms of climatic tables in the Appendix. The follow- ing sections deal with a selection only of important and less well- known characteristics. CHAPTER VI TEMPERATURE VI. 1. HOURS OF OBSERVATION. REDUCTION TO THE "TRUE MEAN" THE INTERNATIONAL REGISTER assumes that the elements, with the exception of precipitation, are observed three times a day. This system is used in most of the European countries, in Russia, from the Baltic Sea to the east Asiatic coast, and in some other networks of climatological stations. The method of three daily observations is therefore important for the climatographies of at least two continents. The observations are so distributed over the day that one is made in the morning, one in the afternoon, and the third in the evening. For an easier understanding, Figure 18 represents a daily course of temperature derived from 50 years 1 thermograph records and related to the middle day of the month. The average tem- 81 82 METHODS IN CLIMATOLOGY perature is characterized by the ordinate of that rectangle which is equal to the area enclosed by the curve (top) , the abscissa (bottom) , in Figure 18 located at 14C, and the limiting ordinates (sides). Generally this area is measured by a planimeter. In the case of a smooth curve, such as that in Figure 18, one can estimate the c 22 20 M 18 16 14 HOURS II N I 3 9 II M .M 1.9 1.6 | I.2|0l8| 0.8(1.5 J2.9M.O 5Op.9 |68|Z5 |a9|e.7|a6 kelrsks |5.4|4.4| 3.TP3 \2.7 AREAS, 0.2 5 SQ. CM t FIG. 18. The daily course of temperature at a place in Central Europe, in August (C) areas between the ordinates of two consecutive hours; the sizes of these areas can be read from the numbers on the J\T-axis, The unit is 0.25 sq cm. The sum of these areas divided by the number of intervals (i.e., 24) gives the height of the average ordinate. In the present case, 4.57C results. Therefore the average temperature required is 4.57 + 14.00 = 18.57C. The method of numerical integration is correct but complicated. TABLE 21. AVERAGE DAILY COURSE OF TEMPERATURE AT A PLACE IN CENTRAL EUROPE (AUGUST) c 6 8 10 11 12 A.M. 16.1 15.8 15.4 15.1 14.8 15.0 16.1 17.4 18.5 19.4 20.3 21.1 P.M. 21.9 22.6 22.7 22.5 21.9 21.0 19.8 18.8 18.1 17.5 17.0 16.6 arithmetic mean 18.56 C TEMPERATURE 83 We can arrive also at a practically identical result by a very simple computation. The arithmetic mean of the 24-hourty temperatures (see Table 21) yields 18.56C. Thus the arithmetic mean of the 24-hourly temperatures is defined as the true mean temperature. 1 This definition is valid for all elements which are continuously recorded or observed at each hour, except vectorial values. We return to the international form of the climatological register and observations at certain fixed hours of the day. The aim of these repeated observations during the day is to learn some- thing about the daily variation of temperature (or another ele- ment), on the one hand; and, on the other hand, to figure out the average temperature of an individual day, and the true means for the months and for the year. Two statistical results are immediately derived from the three (two in some networks) daily observations: (1) The average of the daily observations is defined as the daily mean (valid for all ele- ments except wind direction). (2) The temperatures of each ob- servation hour are averaged over the month. Monthly average temperatures for the fixed hours result. A simple combination of these average observational hours should yield the true monthly mean. A difficulty exists in so far as the form of the daily course is variable from region to region, and in the course of the year. Before we continue this discussion, it is necessary to give some numerical examples of observational hours in the United States and elsewhere. The hours which are highly recommended by the International Meteorological Committee 2 and used in many European meteor- ological networks are : 7 A.M., 2 P.M., 9 P.M. 1 The usual nomenclature is true daily mean. This denomination is appropriate as long as 24-hourly temperatures are concerned. Then the expression is valid for any single day as well as for the average of the month. It is another matter with observa- tions at fixed hours. We then have to discriminate between: (a) the daily mean and (b) the mean of the monthly averages of the temperatures at fixed hours. In case (a), we speak only of a daily mean, which will be discussed later on. Only from the monthly average temperatures at the fixed hours can a combined and corrected value be derived which shows an optimum of agreement with the 24-hour mean. This approximation is called true mean in the following discussion. 1 International Meteorological Codex, p. 24. 84 METHODS IN CLIMATOLOGY Other recommended hours are : 6 A.M., 2 P.M., 10 P.M. 7 A.M., 2 P.M., 10 P.M. 7 A.M., 1 P.M., 9 P.M. 8 A.M., 2 P.M., 8 P.M. 3 In the United States the hours of observations shown in Table 22 were used. 4 TABLE 22. OBSERVATION-HOURS IN THE UNITED STATES (AFTER FRANK BIGELOW) 1870, Nov. to 1872, Aug. 24, 7.35 A. M. 4.35 P.M., 11.35 P.M. WMT> 1872, Aug. 24, to 1879, Oct. 31, 7.35 A. M. 4.35 P.M., 11.00 P.M. WMT 1879, Nov. 1, to 1886, Dec. 31, 7.00 A. M. 3.00 P.M., 11.00 P.M. WMT 1887, Jan. 1, to 1888, June 30, 7.00 A. M. 3.00 P.M., 10.00 P.M. 75th MT 1888, July 1, till about 1930 8.00 A M. 8.00 P.M. 75th MT 1930 to 1936, 7,00 A. M. 7.00 P.M. From 1936, 7.00 A. M. 1.00 P.M., 7.00 P.M., 1.00 A.M. Dr. C. F. Brooks has been so kind as to bring down to date Bigelow's data. 6 WMT = Mean time of the meridian of the observatory at Washington, D. C. 75th MT = Mean time of the meridian 75W = Eastern Standard Time. WMT - 75th MT 8.3 minutes. JP Table 23 contains examples of the annual course of the differ- ence between the temperature obtained from the combination of the observational hours (t ) and the true mean (derived from 24- hourly observations, (/*). The difference c = t t t is called correction to the "true mean.'* It is t t = t + c This means that the correction c is to be added to t without changing its sign. 1 There is also a series of only two observational hours, recommended by the IMC: 8 A.M., 8 P.M. 9 A.M., 9 P.M. 10 A.M., 10 P.M. The averages of the temperatures at these hours do not show great differences from the true mean but do not indicate anything about the daily variation of temperature. 4 According to the investigations in this problem by Frank H. Bigelow, " Report on the Temperatures and Vapor Tensions of the United States," U. S. Dept. of Agriculture, Weather Bureau, Bulletin S (Washington, 1909). See also Alexander McAdie, Mean Temperatures and Their Corrections in the United States (U. S. War Dept., Washington, 1891). TEMPERATURE 85 TABLE 23. THE ANNUAL COURSE OF "c" (F) FOR DIFFERENT HOURS OF OBSERVATION Buffalo, N. Y. (42.9N, 78.9W, 770 ft.) Central Europe Hours 7A.M. 3 P.M. 11 P.M. 8 A.M. 8 P.M. max. min. 7 A.M. 2 P.M. 9 P.M. 7 A.M. 2 P.M. 2X9 P.M. Jan. 0.0 + 0.4 +0.4 -0.9 +0.4 Feb. 0.0 +0.6 +0.4 -0.5 -0.2 Mar. +0.1 +0.8 +0.2 -0.3 +0.1 Apr. +0.2 +0.5 -0.2 -0.4 -0.1 May +0.3 +0.2 +0.1 -0.7 -0.3 June +0.3 0.0 +0.3 -0.9 -0.5 July +0.3 0.0 +0.3 -0.9 -0.5 Aug. +0.3 +0.4 +0.3 -0.1 +0.3 Sept. +0.4 +0.8 +0.4 -0.1 +0.3 Oct. +0.3 + 1.2 +0.5 -0.4 0.0 Nov. +0.1 +0.6 +0.3 -0.8 -0.3 Dec. 0.0 +0.4 +0.5 -1.0 -0.7 Year +0.2 +0.5 +0.3 -0.6 -0.2 X\c\ 2.3 5.9 3.9 7.0 3.7 AR 0<4 1.2 0.7 0.9 1.1 ARx2\c\ 0.9 7.1 2.7 6.3 4.1 The first column shows c for three daily observations, the second for two, the third for the extremes, the fourth for three other hours, and the fifth offers a new feature. Here the observations at the three different hours are not combined in a common arithmetic mean, but in a weighted mean. Double weight is given to the average temperature at 9 P.M. Later on, we shall discuss this method. The question now to be answered is, which hours and which combinations of observations are most advantageous. There are two requirements: 1) The sum of the absolute values of the monthly correc- tions should be as small as possible. 2) The annual variation of the corrections should be as small as possible. The two requirements can be examined by simple criteria as follows: the third line from the bottom in Table 23 shows the sums of the monthly corrections, disregarding the signs (S|c|). It is clear that the first requirement is the better fulfilled the smaller the term S I c \ is. 86 METHODS IN CLIMATOLOGY The second line from the bottom (Table 23) contains the an- nual range (AK) of the corrections, i.e., the difference between the largest and the smallest correction in the course of the year. It is obvious that the product of the two quantities should be as small as possible: AR X S|c| = minimum The hours 7 A.M., 3 P.M., 11 P.M., are indeed by far the best choice, and the hours 8 A.M., 8 P.M., the worst of the five. We see, too, that the result from identical hours of observation can be improved by weighted averages. The third combination (7 A.M., 2 P.M., 9 P.M.) is only slightly better than the second) (8 A.M. + 8 P.M.). The weighted mean is much better than either. (See last line of Table 23.) Nevertheless, three observations a day are much more advantageous, since they give rather good information about the daily variation of temperature. That is not the case at 8 A.M. and 8 P.M., because these times are close to the hours of the daily mean. The difference between the average of the hours of observations and the true mean can be decreased by giving different weight to the observations as has been shown in one special example. If we assume three observations a day, then generally, the following equation holds: I = pta + qtb + rt c if i stands for "true mean," the subscripts a, b, c denote the different hours of observations; the letter /, the average monthly temperature at the respective hours. The letters p, q, r are coeffi- cients which have to be computed from the observations. Ob- viously the condition is valid : p+q+r = 1 If there are complete observations at the fixed hours and syn- chronous, continuous records of temperature, say for two Januaries, the problem is very simple. 6 There are li and 2 from the continuous records and / a ,i, t a ,*, /&,i *&,2, *c,i *c,2 for the average temperatures at the observational hours in the two January months, indicated by the subscripts 1, 2. 9 The once chosen hours and the resulting combination of the monthly mean yields variable differences from the "true mean" in the course of the year. TEMPERATURE 87 Then the following equations are valid : pta.i + 2/6,1 + rt Ct i = li plat + 3^6,2 + r/a,2 = *2 p + q + r = 1 From these three equations, the coefficients p, q, r are computed ; in reality more than 2 years' observations have to be available, in which case the three coefficients can be calculated with the method of least squares. 6 Examples of weighted arithmetic means of the temperatures at fixed hours of observation follow: a) For the greater portion of Europe the observational hours 7 A.M., 2 P.M., 9 P.M., with double weight given the 9 P.M. tempera- ture, are very advantageous: | = (/ 7 + / 2 + 2/ 9 ) : 4 = 0.25 t 7 + 0.25 / 2 + 0.50 / 9 , or p = 0.25, q = 0.25, r = 0.50 b) Subtropics: I = D? + fe + /9 - 1/10 (/i - /Q)]: 3, or = 0.33, q = 0.30, r = 0.37 c) Tropics: I = [2(* 7 + /*) + 3/9]: 7, or p = 0.286, q = 0.286, r = 0.428 The same method is applicable if only the extremes of tempera- ture are observed once a day. This problem is of special importance. About 5500 coopera- tive observers in the United States make but one observation within the 24 hours of a day, and the same method is used in other countries. For reducing the extreme temperatures to the true mean the formula used is: t t = m + k(M - m) For further information see Nils Ekholm, "Reduction of Air Temperatures at Swedish Stations to a True Mean/' MWR, vol. 45 (1917), p. 58; Nils Ekholm, "Calcul de la temperature moyenne de 1'air aux stations mtorologiques Su^doises," Appendicc aux observations mttiorologiques Sutdoises, vol. 56, 1914 (Stockholm 1916). C. E. P. Brooks, "True Mean Temperature," MWR, vol. 49 (1921), p. 226. For instruction for the application of the method of "least squares," see, for ex- ample, H. Arkin and R. R. Colton, An Outline of Statistical Method* (4th ed., New York, 1939). 88 METHODS IN CLIMATOLOGY where M and m stand for maximum and minimum, and k is a factor which is more or less constant for a rather large region and is valid for a certain month ; k varies in the course of the year. C. E. P. Brooks 7 studied this problem in a general manner by calculating a correction term which reduces the arithmetic mean of M and m to the true mean. His formula reads: where a, b, c, are coefficients and R = M m. The correction term depends principally upon the average aperiodical range of temperature (R). It is variable in the course of the year because R has an accentuated annual course at least within the continents. The problem is complicated, furthermore, because the coeffi- cients a, b, c, are functions of the altitude (h) of the place above sea level. It is a = - 0.30 + 0.14 h b = 0.00 - 0.07 h c = - 0.0034 + 0.006 h The elevation h is expressed in kilometers, and the temperatures are absolute temperatures, i.e., T = tC + 273C. Example of a computation of the correction term: Blue Hill (42.2N. 71.1W, 640 ft.) July: M = 79F, m = 61F, h = 640 ft. converted into absolute temperature and kilometer respectively : M = 299 abs., m = 289 abs., h = 0.2 km Therefore: a = - 0.27, b = - 0.01, c = - 0.0022, R = 10C Correction term = a + bR + = - 0.272 - 0.140 - 0.220 = - 0.632 The true mean is therefore 0.6 C or 1.1 F lower than the arith- metic mean of the extremes at Blue Hill, Mass., in the July average. Bigelow determined the difference in question empirically and got 0.5 F. Thus the agreement is not very good as far as the example at random is concerned. 7 C. E. P. Brooks, "The Reduction of Temperature Observations to Mean of 24 Hours, and the Elucidation of the Diurnal Variation in the Continent of Africa/ 1 Quarterly Journal of the Royal Meteorological Society, London, vol. 43 (1917), pp. 375-387, TEMPERATURE 89 The corrections derived from the formula mentioned above are tabulated in Table 24 for different altitudes between km and 3.5 km and for different average aperiodical ranges between 4C and 20C. TABLE 24. CORRECTION REQUIRED TO REDUCE THE MEAN OF THE EXTREMES (M t m). DIFFERENT AVERAGE NON-PERIODIC DAILY RANGES R (C) = M - m AND DIFFERENT ELEVATIONS (h t METERS) ARE THE ENTRIES. (AFTER C. E. P. BROOKS) K(C)\ 500 1000 1500 2000 2500 3000 3500 4 -0.35 -0.42 -0.49 -0.56 -0.63 -0.70 -0.77 -0.84 6 -0.42 -0.50 -0.58 -0.66 -0.74 -0.82 -0.90 -0.98 8 -0.52 -0.59 -0.65 -0.72 -0.78 -0.85 -0.91 -0.98 10 -0.61 -0.67 -0.70 -0.73 -0.76 -0.79 -0.82 -0.85 12 -0.79 -0.76 -0.73 -0.70 -0.67 -0.64 -0.61 -0.58 14 -0.97 -0.85 -0.73 -0.61 -0.49 -0.37 -0.25 -0.13 16 -1.17 -0.94 -0.71 -0.48 -0.25 -0.02 +0.21 +0.44 18 -1.40 -1.04 -0.68 -0.32 +0.04 +0.40 +0.76 + 1.13 20 -1.66 -1.14 -0.62 -0.10 +0.42 +0.94 + 1.46 + 1.98 Frank H. Bigelow made another step in his investigation. 8 When he had computed the annual course of the correction of (M + m)/2 to obtain the true mean, he represented the monthly values cartographically and gave isolines of equal corrections for the United States. The result is the exact and concise summary of an immense numerical investigation which is so instructive that at least the two maps of January and July may be offered here. (Figs. 19 and 20.) The maps show that the trend of the average daily course of temperature is variable from region to region over the continent, as do also the corrections, which give the true mean. It must be emphasized that these variations are not so insignificant. In January they reach 1 F and in July, 2 F. For further purposes, it is important to remember this fact: Mean temperatures derived from extremes without corrections can show differences up to 2 Fahrenheit degrees at places in the United States, even if the true means for an individual month were identical. Some observatories and meteorological stations attempt to make more than three observations between dawn and dusk in order to get the closest approximation to the true mean. From a 8 United States Department of Agriculture, Weather Bureau, Bulletin S (Washing- ton, 1909). 90 METHODS IN CLIMATOLOGY FIG. 19. Correction (F) to give the "true mean" if (M + m)/2 is known. Isolines for the United States, January. (After F. H. Bigelow) FIG. 20. The same as in Fig. 19, but for the month of July TEMPERATURE 91 first glance at the curve of Figure 18 it is clear that rather the contrary is achieved, because the hours above the daily mean get more and more preponderance. If the hours of observation are changed in the course of time, or if the daily extreme temperatures are used instead of fixed hours of observations, the corrections vary in a discontinuous manner at the dates of changing. This can be a serious source of error, for instance in calculating periodicities. The corrections to the plain arithmetic mean vary up to a maximum of about 3/2 F in the United States, in July, if the hours 7 A.M., 3 P.M., 11 P.M. are used, and reach about 2 F in the case of the arithmetic mean of the extremes. Therefore, the climatolo- gist who does not know whether the arithmetic means are weighted, corrected, etc., cannot compare the temperatures at different places with one another; or, if he does so, he should know that the accuracy of the comparison is less than perhaps 2 F. This reason alone makes it uneconomic to aggravate climatological computations, tables, and graphs, with decimals of degrees Fahren- heit. One exception may be a daily or annual course of tempera- ture derived from some years of observations. To avoid misunderstandings, let us repeat that all that has been told about the different corrections of the arithmetic means of temperatures at fixed hours of observations, etc., is valid only for averages of at least ten days. Generally these conclusions are applied only to the observation series of a month and longer periods. The average temperature of an individual day is defined as the arithmetic mean of the temperatures at the observation hours, or of the extremes. These averages can differ from the 24 hours' mean of the day by a considerable amount. The foregoing discussion on the reduction of the arithmetic mean of the records at fixed hours of observation to the true mean is related to the readings of the dry-bulb thermometer. As far as the other climatological elements are concerned, such as vapor pressure, relative humidity, cloudiness, wind velocity, etc., it is usual to identify the arithmetic mean with the true mean. This is the common practice, but it does not claim that there would not be a better approximation to the true mean. For the greater part of the elements, investigations are lacking in this direction. 9 9 The records of cloudiness at the hours of observation, 7 A.M., 2 P.M. (1 P.M.), 9 P.M., are examined because of the differences from the true mean. (See V. Conrad, Met. Zeit. t 1928, p. 23.) It can be shown that the corrections for the above hours are not significant and can be neglected. 92 METHODS IN CLIMATOLOGY VI. 2. DATES AT WHICH THE AVERAGE TEMPERATURE CROSSES CERTAIN THRESHOLDS Generally, these dates are taken from graphs (drawn on a large scale) which represent the annual course. As this method is laborious, a simple interpolation is indicated here. The formula is founded on the fact that the trend of the ascending and descending branches of the curve is linear in rough approximation. There- fore, the interpolation method ife no longer applicable near the extremes. UP DOWN FIG. 21. Diagram explaining the interpolation of the dates at which tempera- ture crosses a given threshold upward and downward. (After V. Conrad) The symbol t means the given threshold of temperature, a the monthly average above it, and b that next below it. The letter D is the difference in days between the middles of the months with the average temperature (b or a) and the middle of the consecutive month (see Fig. 21). The symbol d up is the difference in days between the middle of the month below the threshold and the date at which the temperature (/ ) of the threshold occurs. According to G. Crestani ("Alcuni considerazioni sul calcolo della media della velocitA del vento, sull'andamento diurno ed anno della medesima in Italia," Boll, bimens. XLIX, 1930, 10-14), the combination of wind velocities, i'(9 A.M. + 3 P.M. + 9 P.M.), yields corrections to the true mean which are zero in February and November and reach their maximum in May with about 7% of the true mean. On the average for the year, the correction is 4% of the true mean. If the velocity of the wind is only estimated, not instrumentally measured, the arithmetic mean of the above-mentioned observation hours shows negligible corrections to the true mean. Probably the same is valid with three other reasonable hours of observation. TEMPERATURE 93 The symbol ddo is the same for the descending branch of the curve. Then: d up :D = (t - b): (a - 6) and d do :D = (a - t ):(a - b) Hence a b = D a t a It is sufficiently accurate to assume D = 30 days Finally the interpolation formula reads: u P o a b a b As an example, the annual course of temperature at Bismarck, N. D. (Table 25), will serve. TABLE 25. THE ANNUAL COURSE OF TEMPERATURE AT BISMARCK, N. D. (46.8N, 100.6W, 1670 FEET) O F (*) Deviations op F-Jan. Rel. Temp., % Jan. 8.6 -32.0 0.0 Feb. 9.9 -30.7 1.3 2 Mar. 24.3 -16.3 15.7 26 Apr. 43.3 + 2.7 34.7 56 May 54.3 + 13.7 45.7 74 June 64.2 4-23.6 55.6 91 July 70,0 -f29.4 61.4 100 Aug. 67.5 4-26.9 58.9 96 Sept. 57.9 4-17.3 49.3 80 Oct. 44.6 4- 4.0 36.0 59 Nov. 28.0 -12.6 19.4 32 Dec. 14.7 -25.9 6.1 10 Year 40.6 * After R. DeCourcy Ward and C. F. Brooks, "The Climates of North America," Ktfppen-Geiger, Handbuch der Klimatologie, pt. J (Berlin, 1936). The dates when the temperature rises above and drops below 32, 43, 65 should be calculated. The threshold of 32 divides the curve into the parts below and above the freezing point; 43 is 94 METHODS IN CLIMATOLOGY about the temperature limit of germination of seeds, 65 is the assumed temperature limit for artificial heating. - n 32 -24.3 up 43.3-24.3 and dd - 3Q ^ - 32 23 ^*- JU 44.6 -28.0 -" rf up == 12 is to be added to March 16 and d do = 23 to October 16 Thus we see that : The temperature rises above 32: March 28 The temperature drops below 32: November 8 In the same way it follows that : The temperature rises above 43: April 15 The temperature rises above 65: June 19 The temperature drops below 65: August 24 The temperature drops b^low 43: October 19 VI. 3. DURATION OF TEMPERATURES ABOVE AND BELOW CERTAIN THRESHOLDS These data are very characteristic for the annual course of temperature and should not be omitted. If a table is at hand with running numbers of the dates from January first, the calculation is nothing but a simple subtraction. 10 The duration in days for the three thresholds mentioned, at Bis- marck, N. D. (see VI, 2), is: above 32F: 225 days = 32 weeks = 62% of the year above 43F: 187 days = 27 weeks = 51% of the year above 65F: 66 days = 9 weeks = 18% of the year Sometimes the conversion of the number of days into weeks can be recommended, if the observations are not very accurate, as also when a quick and easy survey is given. This purpose can be better attained with the expression that yields the smaller numbers. Often also the knowledge of the percentage of the year with tem- peratures above and below the thresholds is useful. 10 See Annex IV. TEMPERATURE 95 It is instructive to learn that at Bismarck, N. D., the average temperature remains below the freezing point for 38% of the year and above for 62%, or that only 18% of the year has a temperature above the technically assumed threshold for artificial heating. The duration of a temperature higher than 43 F is often used in climatological investigations and is called the vegetative period. In the United States the term growing season is more or less identical with "vegetative period. " It means the space of time between the last and the first killing frost, or, in the absence of satisfactory frost data, between the last and first minima of 32F or lower. Because I could not find a proper definition of "killing frost" in the available literature, Dr. C. F. Brooks was kind enough to give me the following information. I quote: "A killing frost is one which kills the general vegetation at a place. There is no specific meteorological definition. Where the vegetation is succu- lent and easily damaged by frost, a killing frost will occur at a higher temperature than where the vegetation is hardy. In the absence of killable vegetation, a temperature of 32F in a thermom- eter shelter is taken as a killing frost. Generally, however, the killing frost, first in fall and last in spring, occurs with a minimum shelter temperature higher than 32, sometimes as high as 40F." Therefore, "killing frost" and "growing season" are generally defined biologically, not quantitatively. 11 It goes without saying that the principle of duration is ap- plicable also to many other climatological elements, as average duration of a certain wind direction, of the rainy or hot period of the year (see Figs. 40-41), of the snow-cover, of any average state of the atmosphere or of the surface of the ground, as far as period- ical climatological phenomena are concerned. In all these cases, any information about the respective durations is a good contribu- tion to the knowledge of the phenomenon. VI. 4. OTHER CHARACTERISTICS OF TEMPERATURE CONDITIONS. SPELLS OF COLD AND HOT DAYS Certain thresholds and definitions must be mentioned. In order to discriminate between (a) ice days and (b) frost days, "ice days" are defined as the days with the maximum below the freez- ing point, while "frost days" have the minimum below the freezing point. "See also W. G. Reed, "Frost and the Growing Season," in Atlas of American Agriculture (U. S. Dept. Agr., Washington, D. C., 1918). 96 METHODS IN CLIMATOLOGY Days with a maximum ^ 77F are called (c) summer days, and days with a maximum ^ 86F are called (d) tropical days. All these thresholds are arbitrary. But they are used fre- quently and should be chosen in any case; others can be added which appear advantageous for special purposes. For such thresh- olds, frequencies should be computed. On the other hand, the thresholds are used for calculating spells of frost days, ice days, summer days, tropical days, etc. If some years' observations are at hand, the lengths of the respective spells are evaluated for each January, February, etc., of the series. Then, the average length and the greatest length of the spells should be given. If we have to handle ice days, summer days, tropical days, the temperature-maximum of the consecutive days is decisive for the length of the spells. In the case of frost days, it is the minimum that is decisive. In other investigations, the daily mean of consecutive days is considered. As far as the technique of counting out the duration of spells is concerned, the following rule is valid: If the spell begins in one month and ends in the next, the whole length in days is added to that month to which the greater part belongs ; if the spell is longer than two consecutive montfis, the two months have to be joined. VI. 5. CUMULATED TEMPERATURES Starting from a certain threshold, the mean daily temperatures above or below it are added together for the single month, for the season, or for the year. If some years' observations are available, the respective sums are averaged for the whole period. Especially in the earlier agricultural, meteorological, and climatological investigations, the mean daily temperatures above 43 F (the temperature of germination as stated earlier) are totaled for the single month or for the growing season, etc. A. Angot calculated totals of daily minima at or below the freezing point. Table 26 gives these sums for the region of western TABLE 26. AVERAGE CUMULATED DAILY MINIMUM TEMPERATURES i 0C FOR PARIS, WINTER 1872/73 TO 1911/12 (AFTER A. ANGOT) Centigrade degrees Oct. Nov. Dec. Jan. Feb. Mar. Apr. May Year 3.3 16.1 51.7 59.8 42.7 21.9 3.0 0.2 198,7 TEMPERATURE 97 Europe which is characterized by a climate of slight continentality. This procedure permits also a quantitative distinction between severe and mild winters, if a classification is made according to the standard deviation of the annual sums of minimum temperatures SOC. (See III, 3.) 12 VI. 6. DEGREE DAYS The threshold for artificial heating is a daily mean of 65 F, according to the experience of heating engineers. If the daily mean drops below this temperature, heating is necessary and the amount of fuel to maintain a comfortable inside temperature is more or less intimately correlated with the number of degrees below 65F. This difference (65F minus each day's mean) is called number of degree days. The degree days are totaled for the month, the season, and the year, counting negative differences as zero. If a series of some years' observation is available, the cumu- lated degree days of the month, etc., are averaged. TABLE 27. DEGREE DAYS (F) AT BOSTON, MASS. (42.4N, 7l.lW, 125 FT.) AND CHICAGO, ILL. (41.9N, 87.6E, 673 FT.) (Boston 1914/15 to 1942/43; Chicago 1871 to 1936) Boston Chicago Jan. 1101 1256 Feb. 1027 1083 Mar. 852 903 Apr. 538 545 May 248 280 June 66 69 July 8 8 Aug. 16 11 Sept. 98 94 Oct. 338 363 Nov. 651 742 Dec. 1000 1112 Year 5943 6466 Table 27 gives the cumulated degree days for Boston, Mass., 13 and Chicago, 111. 14 The numbers for Boston are derived from 28 12 See V. Conrad, " Die klimatologischen Elemente und ihre Abhangigkeit von ter- restrischen Einfliissen, Koppen-Geiger, Handbuchder Klimatologie, vol. I B (Berlin, 1936), p. 112. 18 1 have to thank Mr. Edward Sable of the U. S. Weather Bureau for the Boston data. 14 Taken from Thomas A. Blair, Climatology (New York, 1942), p. 18. 98 METHODS IN CLIMATOLOGY heating seasons (1914/15 to 1942/43), those for Chicago from the period 1871 to 1936. The comparison between the two series may be interesting. 18 M J J S A FIG. 22. Relative temperatures at Bismarck, N. D. VI. 7. RELATIVE TEMPERATURES The comparison of the annual course of temperature at one place with that at another place presents two difficulties: 16 In spite of unequal periods. TEMPERATURE 99 1) The average temperatures of the year are different. 2) The amplitudes (annual ranges) are different. 16 The first inequality is overcome by calculating deviations from the arithmetic mean (annual average). (See Table 25, deviations.) The first and second are eliminated by the method of relative tem- peratures, a concept introduced by W. Koppen. The degrees by which the consecutive monthly averages exceed the temperature of the coldest month are each expressed in per cent of the difference between the average temperatures of the warmest and coldest months. By this explanation, the calculation of the numbers of the last column in Table 25 is easily understood. The author has given a graphical representation of relative temperatures 17 which facilitates the comparison of different series of relative temperatures. These are plotted at equal intervals to the right for the first seven months (Fig. 22). Starting from July, the curve turns back to the left, showing the values of the other months. Every point bears the initial of the month. Furthermore, the curve of relative temperatures for January through July is a full line, while the curve of the other half of the year is represented by a broken line. The asymmetry of the trend of the annual course of temperature at Bismarck, N. D., is evident. The months after July are warmer than the symmetrical ones before July. August is warmer than June, September warmer than May, and so forth. In this and in other respects, the representation in Figure 22 gives a good analysis of the annual curve, free from the embarrassing influence of the annual average and of the size of annual variation. 16 The reader should distinguish between annual range and amplitude. The latter is half of the periodical range. 17 V. Conrad, Fundamentals of Physical Climatology, p. 28. CHAPTER VII ATMOSPHERIC PRESSURE AND PRESSURE OF WATER VAPOR PRESSURES are statistically discussed in the same way as tem- perature. It should be emphasized that water-vapor pres- sure is a unilaterally limited element, and there are vast regions where the vapor pressure is practically zero in winter. It is not the aim of this book to deal with the different physical definitions regarding the water-vapor content of the atmosphere. For this, see V. Conrad, Fundamentals of Physical Climatology, and for a more detailed treatment, Sir Napier Shaw, Manual of Meteorology. 100 CHAPTERlVIII WIND VIII. 1. RESULTANT WIND VELOCITY CALCULATED FROM RECORDS OF DIRECTION AND VELOCITY THE RECORDS of an anemograph yield an average direction of the wind for each hour, and the run of the wind, in respec- tive units. The example, Table 28, shows the statistics derived from con- tinuous wind records at Batavia, Java, in January 1926. The first line F gives the number of hours with the various directions TABLE 28. EXAMPLE OF THE CALCULATION OF THE WIND-ELEMENTS; BATAVIA JAVA (6.2S, 106.8E, 26 FT.), JANUARY 1926* N NE E SE s sw w NW I Number of hours (F) 153 37 1 8 41 25 92 137 II Run of wind 309 56 (1) 4 29 29 176 465 (miles) (L) III Average velocity 2.01 1.52 (1.34) 0.56 0.72 1.16 1.92 3.40 (0 = L/F) (mi/h) * Small differences of against the ratios L/F originate from the conversion of meters per second (m/s) into mi/h. during the period in question. The second line shows the run of the wind L for each direction during the same period (here the month of January). In the third line, the ratio L/F appears, i.e., the average velocities of the respective directions. The resultant run of the wind is calculated by Lambert's formula. 1 C N = N - 5 + (NE + NW - SW - SE) cos 45 C w = W - E + (NW + SW - NE - SE) cos 45 where CN and Cw are the total run of the wind from north to south (subscript N) and from west to east (subscript W). If places only in temperate latitudes of the northern hemisphere are considered, it is recommended that the positive sign be given to winds from the north and from the west. It is practical and sufficiently exact 1 Auxiliary tables for the use of this formula are given in the Smithsonian Tables. 101 102 METHODS IN CLIMATOLOGY to make the computation with the value cos 45 = 0.7, so that the calculation can be worked out mentally. C N = N - S + 0.7 (NE + NW - SE - SW) C w = W - S + 0.7 (NW + SW - NE - SE) For 16 directions we get: C N = N - S + (NNE + NNW - SSE - SSJF) cos 22 + (NE + NW - SE - 6W) cos 45 + (ENE + WNW - ESE - WSWQ cos 67| CV = W - E + (WWW + WSW - ENE - ESE) cos + (NW + SW - NE - SE) cos 45 + (NNW + SSW - NNE - SSE) cos 67 (The approximations: cos 22^ = 0.9 cos 45 = 0.7 cos 67| = 0.4 are mostly sufficient). Generally, one can be content with 8 directions. If 16 direc- tions are given in the original data,* it is usual to reduce these to 8 directions. The frequencies of the intermediate directions are allotted by 50% to the two nearest main directions. 2 Analogous formulas can be given for the points of the compass. 3 The values L of the numerical example of Table 28 4 calculated with Lambert's formula yield : C N = + 622 miles for January 1926 Cw = + 479 miles for January 1926 Therefore C N /Cw = tana' = + 622/+ 479 = 1.30 or a' = 52, the angle between the north and the west direction 2 Objections to this procedure are more or less justified theoretically, but in practice this method is a sufficient approximation. 8 If a greater accuracy is desired or if more than 8 directions have to be considered, L. W. Pollak's Rechentafeln zur harmonischen Analyse (Leipzig, 1926) facilitate the procedure. The "Handweiser zur harmonischen Analyse," Prager Geophysikalische Studien, vol. II (Prague, 1928), pp. 54-59, contains special instructions for calculating the resultant wind direction. 4 Data taken from Observations Made at the Royal Magnetical and Meteorological Observatory at Batavia (Batavia, 1930), with some handwritten corrections by H. P. Berlage, Jr. WIND 103 (Fig. 23). Because the azimuth of wind direction is usually reckoned in a clockwise direction (north through east, etc.), the true azimuth a = 52 + 270 = 322 N FIG. 23. Graph for evaluating the true azimuth of the resultant wind VIII. 2. RESULTANT WIND DIRECTION CALCULATED FROM FREQUENCIES OF THE DIRECTIONS ALONE It often happens that only the frequencies of the single wind directions are published, while their average velocities are not known. In this case, also, it is possible to estimate the resulting wind direction. The method is based on the assumption that the average velocities are roughly proportional to the frequencies of the direc- tions. 5 The author examined the wind records for 35 months at Batavia, Java. The correlation between frequency of the direc- tions and the velocity is shown in Figure 24. If, as in this random sample, the assumption in question holds, then winds from the * For instance, if west winds are the most frequent winds at a place, they are also the strongest winds. 104 METHODS IN CLIMATOLOGY most frequent directions are also the strongest. Hence, in the first approximation, the frequencies F (first line of Table 28) can serve for L, the run of the wind. If we calculate with the values F of Table 28, we get a = 321 0.5 1.0 1.5 20nvs FIG. 24. Correlation between the frequency of the directions and the average velocity of the wind, demonstrated by means of the wind records of 35 January months at Batavia, Java. (After V. Conrad) * v. instead of the true value, a = 322. In this example the differ- ence is incidentally negligible. 6 VIII. 3. THE RESULTANT RUN OF THE WIND The resultant run of the wind is given by ft = VCV + C w * = A/622 2 + 479 2 = 785 miles for Batavia, Java, January 1926. 8 The calculation which leads to the result given above is as follows: The numbers of hours with the different wind directions (F) (Table 28, first line) are now introduced into Lambert's formulas instead of the values L. Therefore or and finally C N = (153 - 41) + 0.7( 37 + 137 - 8 - 25) + - 210.7 C w ( 92 - 1) + 0.7(137 + 25 - 37 - 8) = + 172.9 tana' - + 210.7/+ 172.9 - 1.2186 a' 50 38' a - 320 38' WIND 105 The average resultant wind velocity v is calculated by dividing 91 by the number of hours with wind. That means that the num- ber of calms is excluded. The sum of the hours with measurable wind velocity is in the example of Table 28 (first line) Z/ 7 = 494 Therefore 785 494 = l.S9mi/h = 0.71 m/s 7 Frequently, Beaufort numbers are published in the year- books and annals; these should be converted into absolute measure, i.e., m/s or miles per hour, etc., for purposes of comparison. Therefore, Table 29 gives equivalents in meters per second, kilometers per hour, and miles per hour. These equivalents were recommended by the meeting of the International Meteorological Committee at Vienna, Austria, in 1927. 8 TABLE 29. EQUIVALENTS OF THE NUMBERS OF BEAUFORT SCALE * Beaufort numbers m/s km/hour miles/hour 0.25 0.9 0.6 1 1.15 4 2.6 2 2.55 9 6 3 4.30 16 10 4 6.40 23 14 5 8.65 31 19 6 11.15 40 25 7 13.85 50 31 8 16.75 60 38 9 19.90 72 44 10 23.35 84 52 11 27.10 98 61 12 >29 >104 >65 * The average values are calculated from the limits in m/s, originally given by the International Meteorological Committee (1927) for each Beaufort degree. In Table 29, the Beaufort no. is equivalent to 0.25 m/s, owing to the fact that the record "0" (calm) includes also the very weak winds. 7 The necessary tables for conversion of the English measure to the cgs-system, respectively centigrades, meters, etc., are given in the Smithsonian Meteorological Tables (Washington, 1918). 8 See also The Meteorological Glossary (Meteorological Office, London, 1939), p. 36, with slightly different values. 106 METHODS IN CLIMATOLOGY VIII. 4. THE STEADINESS OF THE WIND (5) The steadiness of the wind is the ratio of the resultant run of the wind (9fc) to the run of the wind (R), disregarding the direction, multiplied by 100. 5 =100 1 K If during the whole period considered, the wind always blew in one and the same direction, then 8? = R S Ma * = 100% and we have the upper limit of S. If there is no prevailing wind direction and the wind shifts in a quite incidental way and there are a great many wind observations, the sum of all velocities will be zero if their direction is considered. Then 9? = 0, while R has a finite value (because a place with everlasting calm is impossible). Therefore the lower limit S M in = 0% The steadiness of the wind varies between and 1UU%, according to the above definition. In the example of Table 28, the total run of the wind, disregard- ing the direction (sum of second line) R = 1069 miles The resultant run of the wind (considering the direction) is: $ = 785 miles Therefore 70 cr The complete result may be expressed as follows: The rainy mon- soon at Batavia, in January 1926, had an average net velocity of 1.6 miles per hour, an average azimuth of 322 (W S2N) (N 38W) and a steadiness of 73%. These indications characterize the average wind conditions. The great steadiness of the monsoon wind in January is very significant for this phenomenon. CHAPTER IX SOME COMBINED ELEMENTS IX. 1. RELATIVE HUMIDITY (R.H.) A GRAPHICAL correlation between the monthly average rela- tive humidity and the average temperature, called climo- gram, is frequently used. Figure 25 gives an example for Boston, Mass. 1 The method may be valuable for purposes of illustration. In the present example, it might be stated that the 72 % 70 6*8 66 64 20 30 40 50 60 70 F FIG. 25. Climogram of Boston, Mass. (42.4N, 71.1W, 125 feet) months August through December are more humid than the months March through June, at similar temperatures on the New England coast. This kind of representation is, of course, not restricted to relative humidity and temperature. It can be used for every pair of elements which are subject to an annual course and are related to one another. Fine examples are presented by P. Gotz, 2 who correlates in this way, for instance, the red and infrared content of radiation, turbidity factor, etc., with water- vapor pressure. 1 G. H. Noyes, Annual Meteorological Summary with Comparative Data 1942 (U. S. Weather Bureau. Boston, 1943). See also F. Eredia, "Sul 1 umidita relativa in Italia/ 1 Gerland's BeitAge zur Geophysik, vol. 33 (1931), p. 286. * F. W. Paul G5tz, Strahlungsklima von Arosa (Berlin, 1926). 107 108 METHODS IN CLIMATOLOGY IX. 2. EQUIVALENT TEMPERATURE () Equivalent temperature combines temperature with water- vapor pressure. For illustration: the total water- vapor content of one cubic meter could be condensed at constant pressure. By this process a certain amount of heat would be released, with which the dry air could be heated A/ C. 8 If / is the observed air temperature, then the equivalent tem- perature is defined by the equation : = / + A/ For usual climatic temperatures, the difference A/ = ke which means, it is proportional to the water-vapor pressure, e, where k is the coefficient of proportionality. This coefficient is expressed by the following equation, given by F. Linke: R(607 + 0.7080(273 + 0.239 X 13.6 X b X 1000 where R is the gas constant (R = *29.27 gttfri* sec" 2 ), / = tempera- ture in degrees centigrade and b means the air pressure in milli- meters of mercury (Hg). This formula has been evaluated for different pressures and temperatures between b 770 mm Hg and b = 740 mm, within the limits of the temperatures 20C ( 4F) and 30C (86F). The calculation yields the following results: if / = - 20 and b = 770 mm: k = 1.84 / = + 30 and b = 740 mm: k = 2.16 In the case of simple estimations where no real exactness is re- quired, and at medium pressures and temperatures, it can be assumed that k = 2. Thus - / + 2e 8 The total heat content (W) of a cubic meter of moist air is composed of the heat contents of its dry air and its water vapor. The equation reads: where c p means the specific heat of air at constant pressure, p the density, T the absolute temperature, / the absolute humidity in grams per cubic meter, and r the heat of vaporiza- tion. Dividing the total heat content by c p p, we obtain a value of temperature character. In rough approximation, the absolute humidity,/ (g per m 8 ) and the vapor pressure e are more or less equal numbers as far as usual climatic temperatures are concerned. (See F. Linke, Met. Zeit., 1922, pp. 268 ff.) SOME COMBINED ELEMENTS 109 This formula is valid as an approximation only within the limits mentioned and should not be used if these limits are exceeded by temperature and pressure. As an example, the annual course of the average equivalent temperatures at Boston, Mass., is given (Table 30, columns 5 and 6). The other columns present the numerical values for calculating the equivalent temperature. Finally, in some cases it is desirable to figure out exact values of the equivalent temperature. F. Linke 4 has given another equation : = t + RH-K(t,b) where _ 1543 + 1.68< K(t ' b > " b - 0.377E b and RH = e/E. For evaluating the formula, the temperature / in centigrade degrees, the air pressure in millimeters of mercury, and the rela- tive humidity have to be known. The table in Appendix III is a good expedient, for this computation and yields directly the quan- tity of K(t, b}. The vertical entry is the temperature; the hori- zontal, the air pressure. Examples follow: 1) / = - 18C; b = 780 mm; RH = 0.70 from the table: K = 1.87 e = - 18 + 0.70 X 1.87 = - 16.7C 2) t = 0C; b = 760 mm; RH = 0.70 from the table: K = 9.32 = + 0.7 X 9.32 = 6.5C 3) / = 30C; b = 740 mm; RH = 0.70 from the table: K = 65.2 = 30 + 0.7 X 65.2 = 75.6C The examples show the simple procedure of calculation by means of the table and the interesting effectiveness of vapor content upon the equivalent temperature at different air temperatures. The calculation is still simpler if the temperature of the wet bulb /' is known. Then = /' + K(t' 9 760) 4 F. Linke, Meteor ologisches Taschenbuch (Leipzig, 1931), p. 287. no METHODS IN CLIMATOLOGY Example: if t' - 25.9C, = 75.7C This wet-bulb temperature corresponds to the temperature- humidity conditions of example no. 3. The two calculations of this example yield identical results. In the last equation, corre- lating /' with , one needs consider neither the actual air pressure nor the actual relative humidity. TABLE 30. AVERAGE MONTHLY EQUIVALENT TEMPERATURES (0 = * + 2e) AT BOSTON, MASS. F (1) C (2) e, mm (3) 2e (4) c (5) o F (6) Tan 27.9 2.3 2.6 5.2 2.9 37 Feb 28.8 1.8 2.6 5.2 3.4 38 Mar 35.6 2.0 3.4 6.8 8.8 48 Apr 46.4 8.0 5.2 10.4 18.4 65 May 57.1 13.9 7.9 15.8 29.7 85 June 66.5 19.2 11.2 22.4 41.6 107 July 71.7 22.1 13.6 27.2 49.3 121 Aue 69.9 21.1 13.3 26.6 47.7 118 Sent 63.2 17.3 10.5 21.0 38.3 101 Oct 53.6 12.0 7.2 14.4 26.4 80 Nov 42.0 5.6 ,4.8 9.6 15.2 59 Dec 32.5 0.3 3.2 6.4 6.7 44 Column 1 : Air- temperature in F. Column 2: Air- temperature in C. Column 3: Water-vapor pressure in millimeters of mercury. Column 4: The values of Col. 3 doubled. Column 5: Equivalent temperature in C. Column 6: Equivalent temperature in F. TABLE 30a. THE EQUIVALENT TEMPERATURES COMPUTED FROM 8 = t + 2e (TABLE 30), COMPARED WITH THE RESULTS (G') FROM THE CORRECT FORMULA (SEE p. 109).* c RH k(t. 760) KXRH e /o c ec <',c e) I -2.3 0.67 7.83 5.2 2.9 2.9 0.0 II -1.8 .64 8.13 5.2 3.4 3.4 0.0 III 2.0 .65 10.7 7.0 9.0 8.8 +0.2 IV 8.0 .65 16.2 10.5 18.5 18.4 +0.1 V 13.9 .66 24.0 15.8 29.7 29.7 0.0 VI 19.2 .67 33.4 22.4 41.6 41.6 0.0 VII 22.1 .68 39.7 27.0 49.1 49.3 -0.2 VIII 21.1 .71 37.4 26.6 47.7 47.7 -0.0 IX 17.3 .72 29.6 21.3 38.6 38.3 +0.3 X 12.0 .67 21.1 14.1 26.1 26.4 -0.3 XI 5.6 .70 13.8 9.7 15.3 15.2 +0.1 XII 0.3 .67 9.53 6.4 6.7 6.7 0.0 * K(t t b) is taken from Appendix HI, assuming b the year. 760 mm Hg, constant through SOME COMBINED ELEMENTS 111 IX. 3. DRYING POWER Drying power combines saturation pressure for a given actual or average temperature, water-vapor pressure, and wind velocity. A short instruction on how to calculate this combined element 5 may suffice. The theoretical basis could be improved. The idea is extremely useful in the description of a climate. The definition reads: Drying Power is the amount of evaporation, in centimeters, within four hours. It is calculated by means of the following empirical formula: S 9 = 0.023 X F(v) - X (1 + 0.084 v) e A/ where S v = drying power at the wind velocity, v, in km/hour e = the observed water- vapor pressure in mm Hg E t = saturation pressure at the temperature, tC F(v) = a factor dependent on v 6 AJ3/A/ means the variation of the saturation pressure with tem- perature and can be taken from any psychrometric table which indicates temperature in C and the saturation pressure in mm Hg. Example: The saturation pressure, E, is taken at one degree above and at one degree below the given air temperature. If t = 10C then from the table: E 9 * = 8.61 mm and OQA *u 9,84 - 8.61 = 9.84 mm then - = ~ - - = 0.615 A/ 11 y If AE/A/ is known, the computation of the formula of the dry- ing power offers no further difficulty or doubt. 1 W. Knoche, " El ' Valor de Desecaci6n ' como factor climatol6gico," Revista ChileHa de Historia y Geografia 1919, nos. 34, 35; W. Knoche, " Der ' Austrocknungswert* [Dry- ing Power] als klimatischer Faktor, 11 Aus dem Archiv der Deutschcn Seewarte, vol. 48 (1929), no. 1. 6 The values of F(v) for different wind velocities are: km/hour F(t> km /hour F(v) 1.000 6 1.667 1 1.148 7 1.712 2 1.274 8 1.746 3 1.392 9 1.762 4 1.493 10 1.777 5 1.592 15 1.782 112 METHODS IN CLIMATOLOGY It is only necessary to add that the formula is valid only for sea-level. For other levels, S v has to be multiplied by b /b, where b = 1015 mb and b is the pressure in millibars at the place in question. IX. 4. COOLING POWER Cooling power is observed by the dry and by the wet kata- thermometer invented by Leonard Hill. This book does not deal with instrumental technique, but it may be noted that Hill suc- ceeded so far in analyzing the cooling components a to arrive at analytical equations which give the cooling power from values of temperature, wind velocity, and humidity. Cooling power is a bioclimatological combined element and is defined as follows : The cooling power is equal to the heat loss which a surface of a body at the temperature of the human blood (99F, 37C) experiences, if exposed to the free air. It is measured in milligram calories per square centimeter and second (mgcal/cm 2 , sec). There are two kinds of cooling power: 1) Heat loss of a dry surface (H). 2) Heat loss of a wet surface The equations read : A) for wind velocities less than 1 m/s : dry surface: H = (0.20 + 0.40A/*)(36.5 - /) wet surface: H' = H + (0.060 + 0.072^ 3 )(E - *) 4 ' 3 H = the dry cooling power in mgcal/cm 2 , sec H' = the wet cooling power in mgcal/cm 2 , sec v = the wind velocity in meters per second t = the air tempersture in C E = the saturation pressure in millimeters of mercury 7 e = the vapor pressure in millimeters of mercury B) for wind greater than 1 m/s: dry surface: PI = (.13 + .47Vfl)(36.5 - f) wet surface: H' = H + (.035 + .098z> 1/3 )( - In English units, i.e., when v is the velocity in feet per minute and / the temperature in degrees Fahrenheit, the equations read : *At36.5C (97.7F). SOME COMBINED ELEMENTS 113 dry surface : below 200 ft/min H = (.11111 + .01584^(97.7 - /) above 200 ft/min H = (.07222 + .0186lA/*>)(97.7 - /) wet surface: below 200 ft/min H' = (.19444 + .08 11 8^) (97. 7 - /') above 200 ft/min H' = (.05556 + .10505^)(97.7 - /') The formulas for wind velocities greater than 1 m/s or 200 ft/min seem to hold very well ; but are perhaps not fully valid for the small velocities. 8 The symbol t f of the two equations for the wet surface denotes the temperatures of the wet-bulb thermometer. Newer formulas for the dry cooling power originate from par- ticularly exact experiments: 9 H = 0.99(r(7V - TV) + (.113 + .34^- 622 )(^ - T A ) where T K denotes the absolute temperature of the kata-thermom- eter, T A the temperature of the air, and <r the constant of the Stefan- Boltzmann-law : a = 8.26 X lO For moderate conditions, i.e., (T K + T\0:2 = 300a, in very rough approximation, the formula above can be simplified: 10 H = (1 K - t A )(.26 + .34t/>- 622 ) where IK and IA mean centigrade degrees. 8 V. Conrad, "Messung und Berechnung der Abkiihlungsgrosse," Gerland's Beitrdge zur Geophysik, XXI (1929), 183. For further information See D. Hargood Ash and Leonard Hill, "The Kata- thermometer," in Studies of Body Heat and Efficiency (London, 1923); C. F. Brooks, "The Cooling of Man under Various Weather Conditions," MWR, vol. 53 (1925), pp. 423-424; E. C. Donnelly, "Human Comfort as a Basis for Classifying Weather," ibid., pp. 425-426; V. Conrad, Fundamentals of Physical Climatology, where the reader will find a diagram (p. 113) from which he can make rough estimations of dry cooling power for temperatures between 50F and +98F and wind velocities from calm to 25 mi/h. 9 H. Lehmann, Mikroklimatische Untersuchungen der Abkuhlungsgrosse in einem Waldgebiet (Leipzig, 1936). See also R. G. Stone, "On the Practical Evaluation and Interpretation of the Cooling Power in Bioclimatology," BAMS, vol. 24 (October 1943), pp. 304 ff . 10 K. BUttner, Physikalische Bioklimatologie (Leipzig, 1938), p. 98. 114 METHODS IN CLIMATOLOGY For bioclimatological purposes IK has to be assumed as con- stant at 36.5C. It should also be emphasized, from the methodo- logical standpoint, that the relations of cooling power to physio- logical processes are very complicated. The application of cooling power to the problems of the heat balance n of the human body assumes that the temperature of the human skin is constant. This assumption does not hold. 11 See C. F. Brooks and E. C. Donnelly, cited in note 7 above, and D. Brunt, in QJRMS, 1943. CHAPTER X CLOUDINESS IN THE FOREGOING DISCUSSION of variates of different kinds, the statistical properties of cloudiness were considered, and their relation to the relative sunshine duration too. (See III. 6 and III. 10.) Especially for this element, average values have to be supplemented by computing frequency distributions. 1 When observing and discussing cloudiness, one should discriminate be- tween low and high clouds. 2 X. 1. CLEAR AND CLOUDY DAYS In a number of national meteorological networks no detailed observations of the cloudy part of the sky are made. Only the day as a whole is characterized ' 'clear/' "cloudy," or "partly cloudy." In the United States, this kind of observation is usual at the cooperative stations, because records are made only once a day. The international definitions (Utrecht, Holland, 1874) are: 1) Clear days are days with an average cloudiness between 0% and 20% of the visible sky. 2) Cloudy days have an average cloudiness between 80% and 100%. The instructions for cooperative observers of the U. S. Weather Bureau which differ from the international use are as follows: 4 'The general character of the day from sunrise to sunset should be recorded as 'clear* when the sky averages three tenths or less ob- scured . . . and 'cloudy' when more than seven tenths obscured." A successful effort has been made to correlate the frequencies of clear and cloudy days with the average percentage of cloudiness. The records at stations where cloudiness as well as frequencies of clear and cloudy days are simultaneously observed, give the op- portunity of establishing an analytical relation between cloudiness 1 W. KOppen, H. Meyer, " Die HSufigkeit der verschiedenen Bew6lkungsgrade als klimatisches Element/' Archiv D. Secwarte, XVI (1893), no. 5. * A. AngstrOm, "Ueber die Schatzung der BewSikung," Met. Zeit. (1919), pp. 257- 262. 115 116 METHODS IN CLIMATOLOGY and frequency. A likely assumption is that cloudiness is propor- tional to the difference of the frequency of cloudy days minus that of clear days. If o means the average frequency of cloudy days and c that of clear ones, and n denotes the number of days of the month, season, etc., the following equations are valid for different regions : Norway C = 51 + 51 Russian Empire C = 50 + 52 Palermo, Sicily C = 48 + 40 n o c n o c n In Spain, the same definitions are used as in the United States. The formula reads: Spain C = SO + - (o - c) TABLE 31. FREQUENCIES OF CLEAR (c) DAYS AND CLOUDY (o) DAYS AND THEIR RELATION TO CLOUDINESS (C%) (Example taken from the Observations at Blue Hill Meteorological Observatory) Month -C Days Cloudiness Obs. Calc. % % Obs. - Calc. % Jan 4 56 52 56 53 57 56 53 52 49 50 54 54 55 51 57 55 57 57 51 51 49 49 55 55 + 1 + 1 -1 -2 -1 +2 + 1 + 1 -1 -1 Feb 2 Mar 5 Apr 4 May 5 June 5 July 2 Aue 2 Sept 1 Oct 1 Nov 4 Dec 4 Year 3 53 53 2(+) = 6 Z(-) = 6 CLOUDINESS 117 For Blue Hill Observatory, south of Boston, Mass. (42.2N, 71.1W, 640 ft) simultaneous observations of cloudiness and fre- quency (c and 0) are available. 3 With the assumption that C = a + b (o c), one obtains the equation C = 47 + 2 (o c}. This equation differs from those previously mentioned in that no distinction is made between the different lengths of the months. This may be an example of avoiding an unfounded and exaggerated accuracy. The numbers in Table 31 show that the calculated values are in very good agreement with those observed, in spite of the simplified equation. X. 2. FOG The usual statistical characteristics are : 1) Average number of days with fog, divided sometimes into different degrees of density. 2) Probability of a foggy day ( = the number of foggy days divided by the number of days in the month). 3) Average duration of fog, in hours, on a foggy day. The last characteristic is computed by the method of random samples. The procedure is identical with that of the duration of precipitation which was discussed earlier (III. 10). Usually more detailed statistics are compiled only for the oceans. Six observa- tions every 24 hours are made at sea, so that in this case the method of random samples is very successful. * A. McAdie, "Observations and Investigations made at Blue Hill Meteorological Observatory, Mass., 1915 with Summaries for Thirty Years 1886-1915," Annals of the Astronomical Observatory of Harvard, vol. LXXIII, pt. Ill, (Cambridge, 1916). i CHAPTER XI PRECIPITATION F THE PRECIPITATION is measured with the standard rain- gauge, not with a self-recording instrument, two primitive elements can be taken directly from the records: 1) Number of days with measurable precipitation 1 la) with rain Ib) with snow Ic) with hail Id) with sleet 2) Total amounts of precipitation fallen during the single days, during a month, etc. Starting from these data one proceeds to the following concep- tions. XL 1. PROBABILITY (P) OF *\ DAY WITH PRECIPITATION This is the quotient of the number of days with precipitation (r) over the total number of days of the period in question month, season, year (n) : P = 100 - % n (Boston, Mass., has 125 days with precipitation in the average of 60 years). Therefore P = 100 = 34.2% = 342/1000 1 Recently H. Neuberger ("On the Measurement and Frequencies of Traces of Precipitation/' BAMS, vol. 25, 1944, pp. 183-188) has published interesting statistics on traces of precipitation measured by means of new instruments which overcome some difficulties of the instruments of S. P. Fergusson and others. Here, it is not the instru- mental side which is of interest but the fact that the investigations permit differenti- ating frequencies of traces and measurable precipitation. Statistics of the two phenom- ena separated from one another would be valuable and also, e.g., annual courses of the ratio of the frequencies of traces and real precipitation. Not only applied climatology (agricultural) would profit much. Numbers of days with traces mean a new and important precipitation element if the traces are observed in a well-defined way. In any case, this element is at least as consequential as the number of days with dew. 118 PRECIPITATION 119 If only per cent is calculated, it suffices to take the year with 365 days. XL 2. PROBABILITY THAT A DAY WITH PRECIPITATION Is A DAY WITH SNOWFALL Generally, this probability should be calculated only for the average of the snowy period of a climatic region (Table 32), not TABLE 32. PROBABILITY THAT A DAY WITH PRECIPITATION is A DAY WITH SNOW- FALL (Ps). BOSTON, MASS. NUMBER OF DAYS WITH PRECIPITATION AND SNOW- FALL, (60 AND 31 YEARS RESPECTIVELY), TAKEN FROM G. H. NOYES, Meteorological Summary, Comparative Data (U. S. Weather Bureau, Boston, 1943) Month Number of Days with Precipitation Number of Days with Snowfall PS % July 10 Ausr 10 Sept 9 Oct 9 1) (0) Nov 10 3 30 Dec 11 8 73 Jan 12 11 92 Feb 10 10 100 Mar 12 7 58 Aor 11 3 27 May 11 (<1) (0) June 10 for the whole year. If the number of days with precipitation is r , and s of them are days with snowfall, the probability P. = 1005/r% XL 3. AMOUNTS OF PRECIPITATION WITHIN CERTAIN PERIODS The amounts of precipitation for single days are totaled for different sections of the year according to the purposes in question. There are average sums and individual amounts per week, ten days, month, season, year. For agricultural purposes, the amount of precipitation which occurs during the vegetative period (growing season) will be of interest (see the explanation in the chapter on temperature, VI. 3). The beginning and end of these periods is either derived from the annual course of temperature, or from bio- logical and phenomenological experiences; for instance, the time from the average date of sowing to that of the beginning of the flowering season. 120 METHODS IN CLIMATOLOGY In an ordinary climatography, at least the monthly and the annual amounts have to be given. Average amounts of precipita- tion should be given in millimeters; frequently better, in centi- meters ; or else, if possible, in hundredths of inches and otherwise, in tenths. This is the accuracy which corresponds to the exactness of the observations. XL 4. THE ANNUAL COURSE OF THE AMOUNTS OF PRECIPITATION The annual amount of precipitation may be considered as being evenly distributed over all days of the year. Monthly sums can then be calculated. The resulting course shows, however, varia- tions caused by the unequal length of the months. This incon- venience gives rise to different methods for eliminating this source of error. XL 4 a. The Reduction to Months of Equal Length This can be achieved in two ways. The simpler way is to re- duce the monthly amount to equal months of 30 days. The amounts for the months with 31 days are reduced by 3.2%. The amount for February (28.25 days) fe increased by 6.2%. Table 33 TABLE 33. MONTHLY PRECIPITATION (MM) AT A PLACE IN CENTRAL EUROPE REPRESENTED BY DIFFERENT METHODS 1 2 3 4 5 6 7 Col. 2 reduced Even distri- cart Average to equal Col. 2 in bution of pluviomttrique Relative monthly months thousandths of an annual relatif pluvio- precipitation of 30 days the annual amount of (Col. 4 minus metric Month mm mm amount 1000 Col. 5) coefficient Jan 37 36 57 85 -28 0.67 Feb 33 35 51 77 -26 0.66 Mar 47 45 73 85 -12 0.86 Apr 53 53 82 82 1.00 May. . . . 71 69 110 85 25 1.29 June. . . . 70 70 108 82 26 1.32 July 79 76 122 85 37 1.43 Aug.. . . 69 67 106 85 21 1.25 Sept.. . . 50 50 77 82 5 094 Oct 47 45 73 85 -12 0.86 Nov. 45 45 69 82 -13 0.84 Dec 47 45 73 85 -12 0.86 Year. .648 (636) 1000 1000 PRECIPITATION 121 contains in the second column the original monthly sums at a place in Central Europe. They are reduced to months of equal length of 30 days in the third column. The second method is to reduce the original sums (column 2) to a month which has the length of the twelfth of a year. That is 30.438 days. Therefore the amounts for February have to be multiplied by 1.077 for the months with 30 days, by 1.015 for the months with 31 days, by 0.982. XL 4 b. Amount of Precipitation on an Average Day of the Month The monthly amount is divided by the number of days of the month in question. The advantage of this method is that the inequality of the months is eliminated. The great disadvantage however, is, that these numbers are absolutely fictitious. Thus, it was not of much value for F. H. Bigelow 2 to calculate " Daily Normals of Precipitation " for a great number of stations in the United States. Precipitation is always a discontinuous climato- logical element. Every day and everywhere there is always a temperature ; but there are days with precipitation and days with- out. An evenly interpolated distribution gives a wrong picture of the phenomenon. XI. 4 c. cart Pluviomtrique Relatif and the Relative Pluviometric Coefficient A. Angot suggested two other methods of representation, which though not very illuminating have been frequently used in climato- logical papers and books. His Ecart pluviometrique relatif may best be translated relative monthly deviation. In Table 33, column 5, the numbers of an even distribution over the months are shown for an annual amount of 1000. In column 4, the actual distribu- tion per mille is calculated from column 2. In column 6, the differ- ences (column 4 minus column 5) are given. These differences are called icart pluviometrique relatif, or relative monthly deviations. Monthly quotients are calculated with the numbers of column 4 over the homologous numbers in column 5. These ratios are called 2 Frank H. Bigelow, "The Daily Normal Temperature and the Daily Normal Pre- cipitation of the United States/' U. S. Weather Bureau, Bulletin R (Washington, D. C., 1904). 122 METHODS IN CLIMATOLOGY relative pluviometric coefficients (column 7). 3 Both expressions are scientifically well founded and eliminate the inequalities of the months. All in all, the reduction to equal months of 30 days seems to be the best expedient. The small difference between the actual annual sum and the reduced sum (see last line of columns 2 and 3) is of no significance. XL 5. ABSOLUTE AND RELATIVE VARIABILITY. QUOTIENT OF VARIATION. METHOD OF RANDOM SAMPLES APPLIED TO RECORDS OF RAINFALL 4 These three topics have been discussed in connection with related problems of mathematical statistics. (See III. 9 and III. 10). XI. 6. RAIN INTENSITY (I) The ratio between the total amount during a given period (A) and the number (N) of days with precipitation is called rain in- tensity. It is the average amount of rain of one rainy day. ~N In a 50-60 years' average for August, the amount of rain at Boston, Mass., is A = 3.62 inches, and the number of rainfall days is N = 10. Therefore the rain intensity of August at Boston is / = 3.62/10 = 0.36 inches = 9 mm per rainy day. The corre- sponding data for Cherrapunji in the Khasi Hills in British India (25.3N. 91.8E), in June, the month of maximum, are: A = 2632 mm; N = 24.8; / = 106mm per rainy day. Cherrapunji is located in one of the rainiest regions of the world. There rain intensity in the wettest month is twelve times greater than at Boston. Rain intensity is a very significant characteristic of a climate. 1 Dr. C. F. Brooks kindly draws my attention to two papers of B. C. Wallis, "The Rainfall of the Northeastern United States" and "The Distribution of the Rainfall in the Eastern United States," MWR, vol. 43 (1915), pp. 11-14 and 14-24, where the pluviometric coefficient is applied to the rainfall in the eastern United States. 4 Following a British custom, rainfall, being the shorter expression, is often used instead of precipitation. PRECIPITATION 123 XL 7. WET AND DRY SPELLS The average and extreme lengths of wet and dry spells are of decisive influence upon organisms. A wet spell is defined as a period of consecutive days with at least 0.01, 0.05, etc., inches, or 0.1, 0.2, 0.5, etc., millimeters of rain. The thresholds on which the definition is founded have to be expressly indicated because no general agreement exists in this respect. It is assumed that about ten years' records are at hand. This is sufficient. A much shorter period should not be chosen. If the ten years are available, the length of the spells (wet or dry periods) should be excerpted from the monthly records. An isolated rainy day means a spell of one day for statistical reasons. Then a) the average length of a wet (dry) spell b) the average length of the longest wet (dry) spells c) the extreme lengths are computed for each month. If a spell overlaps from one month to the next, or if its length is greater than one month, the rules for hot and cold spells govern (see VI. 4). A dry spell means a period of at least five consecutive rainless days. It goes without saying that the establishment of a minimum limit is necessary; a shorter period is unlikely to damage the vegetation in any way. "Dry periods 1 ' longer than five days are not considered as being interrupted by a rain less than 1mm 0.04 inch). On the coast of the middle Adriatic Sea, dry spells up to the extreme of 81 days are recorded in summer. The longest rainy period observed in the eastern Alps within 40 years had a duration of 16 days (in September). These two examples serve to illus- trate dry and wet spells. The definitions mentioned are by Julius Hann, slightly modified by the author. It is regrettable that there is no international definition of wet and dry spells. In the British Meteorological Service, other definitions are used : 6 Drought is dryness due to lack of rain. Certain definitions have been adopted in order to obtain comparable statistical information on the subject of droughts. An absolute drought is a period of at least 15 con- 1 The Meteorological Glossary (3rd ed., Meteor. Off. London, 1939). See pp. 68, 161, 212. 124 METHODS IN CLIMATOLOGY secutive days, to none of which is credited 0.01 in. of rain or more. A partial drought is a period of at least 29 consecutive days, the mean daily rainfall of which does not exceed 0.01 in.; (the word mean is to be em- phasized). A dry spell is a period of at least 15 consecutive days to none of which is credited 0.04 in. rain or more. Rain spell is a period of at least 15 consecutive days to each of which is credited 0.01 in. rain or more; thus the definition of the term rain spell is analogous to that of the term " absolute drought. Wet spell. Analogous to the term dry spell, the wet spell is a period of at least 15 consecutive days to each of which is credited 0.04 in. of rain or more/' These definitions are surely well fitted to the climate of the British Isles. The definitions by Hann-Conrad have perhaps more gen- eral applicability and are less complicated. The American definition of drought reads: " period of 20 (or 30) consecutive days or more without 0.25 inch of precipitation in 24 hours during the season, March to September, inclusive." XL 8. CHARACTERIZATION OF THE HYDROMETEORIC TYPE OF A PERIOD Frequently it is necessary to give* a general characterization of a week, month, year, etc., as far as rainfall is concerned. The ques- tion as to whether a month is to be called dry, for instance, or extremely dry, wet or extremely wet, etc., in a given climate, is answered in III. 3. By means of the standard deviation <r, the problem is solved in a well-defined and reproducible way. XL 9. SNOW Table 34 shows a form representing snow conditions which has proved to be advantageous. The form serves for a single year as well as for a series of years. In the latter case, averages replace the individual values. If possible, extreme values taken from the period in question should be inserted; e.g., between columns 6 and 7, the earliest and the latest date of both the beginning of the first and the ending of the last snow-cover. This makes 4 columns more, with increased inconvenience for printing. A supplemen- tary table of extremes is often more convenient. In Column 7 is shown the number of days on which a snow- cover actually exists. PRECIPITATION 125 In contradistinction to this characteristic, the season of snow- cover has to be indicated in Column 8. The season means the number of days between the date of the beginning of the first snow-cover and that of the end of the last one. Within the "sea- TABLE 34. FORM FOR TABULATING SNOW CONDITIONS FOR A NUMBER OF STATIONS Period to ( years) [Remarks regarding observation-hour and methods] 10 11 12 Date of the 0) ^ S 5 3 M s * M * 1 i 1 J I CU -o JS M CO CO -S S c co ^ i CO 1 I 3 5 1 IS 1 is Beginning 0) -o W CO 1 "o 8 CO c CO JC CO 1 If *T3 O O a 0) *0 0) a a Ui O t u C 3 U TD S 2 J c o -Q ^O N ' s.s oJ I i Snow cover S 3 S rt Q 3 II C/J co X _- II son" the snow-cover can melt away and be formed anew. The number of days with an actual snow-cover is called duration of snow- cover, and is always smaller than, or at the most, equal to the season. The greater the elevation and the geographical latitude, the smaller the difference between season and duration. For example, in the eastern Alps at one place at 4000 feet altitude the difference is 20 days, at another place at 700 feet it is 52 days. The data of column 11 can be given only if the freshly fallen snow is regularly measured. These numbers are totaled for the month or the year. The records of the cooperative stations of the U. S. Weather Bureau enable one to compile statistics of the lengths of the single 126 METHODS IN CLIMATOLOGY periods of the snow-cover. This is important especially for regions where the snow surface offers easy transportation, e.g., logging, skiing, etc. The variability of the duration as well as of the season is of im- portance and of interest. It decreases with elevation under other- wise equal conditions. Average values of duration and season have to be reduced to a standard period by the method of differ- ences. Values derived from different periods are not comparable. If Z means the duration in days and 5 the annual amount of melted snow in millimeters or inches, then the ratio Q = Z/S yields the duration value (Q) in days for 1mm (inch) of snow. This expression can be useful in some investigations. The percentage of the total amount of precipitation which falls in the form of snow should be indicated in every regional descrip- tion of snow conditions. In more detailed studies into the depth of snow on the ground, average month or half-month values are calculated. The statistics of longer series of years yield probabilities of certain average depths of snow in certain weeks, half months, etc. 6 In mountainous countries, the depth of snow on the ground is, of course, dependent to a high degree upon altitude. Therefore, also, cartographic representations oT depth, duration, season, etc., can be based only on the principle of isanomals. Further details are beyond the scope of the present work. 7 8 For a good pattern see R. G. Stone, <4 The Distribution of the Average Depth of Snow on Ground in New York and New England: Method of study," Transactions of the Am. Geophysical Union, 19th Annual Meeting (1938), pp. 486-492, and the second part, "Curves of Average Depths and Variability," ibid. (1940), pp. 672-692. 7 For further discussion see " Mitteilungen (iber die Schneedecken-Verhciltnisse in den Ostalpen, 11 in Gerland's Beitrage zur Geophysik, as follows: (1) V. Conrad and M. Winkler, " Beitrage z. Kenntnis d. Schneedecken-Verhaltnisse," vol. 34 (1931), p. 473; (2) V. Conrad, "Die Schneedeckenzeit, ihr Anfangs-und Enddatum in den Ostalpen, " vol. 43 (1934) p. 225; (3) V, Conrad, "Der Anteil des Schnees am Gesamtniederschlag und seine Beziehungen zu den Eiszeiten," vol. 45 (1935), p. 225; (4) F. Steinhauser, "Ueber den Schneeanteil am Gesamtniederschlag im Hochgebirge der Ostalpen," vol. 46 (1936), p. 405; (5) E. Ekhart, "Die Andauer der Schneedecke nach Stufenwerten der SchneehShe," vol. 50 (1937), p. 184; (6) V. Conrad and O. Kubitschek, "Die Veriinder- lichkeit und Machtigkeit der Schneedecke in verschiedenen Seehohen," vol. 51 (1937), p. 100. Additional Remark: According to an investigation in a rather extensive material, the variability of the depth of snow on the ground is so great that a reduction to a given period is not successful. Therefore, the discussion only of synchronous series gives reliable results. PRECIPITATION 127 XI. 10. INDEX OF ARIDITY This concept represents an element combined of hydrometeoric elements and temperature elements. First, E. de Martonne's index may be mentioned (in a modified form) : 7 = r+io where n = the average number of rainy days within the period in question, r = the average rainfall in the same period, t = the average temperature. The three variables are related to a certain month, a season, or a year. Another index of aridity has been established by W. Gorczyn- ski. 8 His aridity coefficient (AC) considers the influence of lati- tude, of the range of temperature, and E. Gherzi's measure of variation of precipitation (M - m)/p. (See III. 9.) The funda- mentals of the AC are, however, not yet so well investigated that its routine use can be recommended. Generally, such indices may be applied only with great caution. Often the results are satisfactory in one region and show contradic- tions in another. Furthermore, the boundary conditions are sometimes not fully considered, so that further investigations appear desirable. The numerical indices calculated from these formulae yield relative values fitted only for comparison of the conditions at different stations. Other similar attempts (viz., continentality, border lines of climatic types) are discussed in part XVII. 1 and 2. XI. 11. RAIN-HISTOGRAM Two methods of graphical representation of the annual course of rainfall are usual : 1) the histogram 2) the smooth curve. Both kinds are known from the representation of the frequency distribution (see Fig. 4). The histogram originates from the idea 8 The Aridity Coefficient and its Application to California (The Scripps Institution of Oceanography, La Jolla, Calif., 1939). 128 METHODS IN CLIMATOLOGY already mentioned that precipitation is a discontinuous element. Consequently, this representation is logical. The disadvantage of the method is the difficulty of drawing, and the inaccuracy result- ing therefrom. On the other hand, nobody can forget or overlook the fact that rainfall is a discontinuous element; altogether the smooth curve is preferable. It is more easily drawn and better comparable with other elements. PART III METHODS OF SPATIAL COMPARISON CHAPTER XII COMPARISON OF OBSERVATIONAL SERIES OF DIFFERENT PLACES. GEOGRAPHICAL DISTRIBUTION OF CLIMATOLOGICAL ELEMENTS XII. 1. UNIFORM TENDENCY OF THE VARIATIONS OF AVERAGE WEATHER OVER LARGE REGIONS. CONSEQUENCES DRAWN FROM THIS TENDENCY FOR COMPARING SERIES OF ELEMENTS OBSERVED SIMULTANEOUSLY AT DIFFERENT PLACES IN THE BEGINNING of the first part of this work, some remarks were made about different kinds of observations of climato- logical elements, as far as they are of methodological interest. It was there demonstrated (I. 3. a.) that a change of instrument, even if it means an important improvement, can cause a break in a climatological series. Another very trivial example may be added here. At a " secu- lar " station, the rain gauge happened to leak after having been used for some years. The hole in the bottom was very tiny in the beginning, so small that it closed by itself when dust and dirt were caught with the rain. Then for a time the gauge was watertight. It began to leak again when the dirt was washed away by heavier rain. The acids contained in the rainwater widened the hole in the course of time till the vessel leaked so much that the observer became aware of the defect. Then the gauge was repaired or exchanged, and the cycle began anew. The lifetime of the gauge was determined by workmanship and material. This is the story of one of the many thousands of rain gauges in the world. And what does it mean for climatological methods? As long as the rain gauge is intact, the annual amounts may fluctuate around a certain mean. This decreases when the gauge leaks, increases a little when the hole is nearly closed by chance, decreases again, and so on. The mean happens to reach its 129 130 METHODS IN CLIMATOLOGY former value in the period after a new gauge is installed. Thus, a false cycle of a semi-periodical variation of rainfall comes into existence. This simple and trivial story is obviously of great importance to the student of periodicities in long climatological series. No doubt in this regard many a scientist has been misled. A series of observations such as the one described, showing variations caused by unnatural influences, is called inhomogeneous. It is too easy to find manifold causes of similar inhomogeneities to make further explanation necessary here (growing trees, new build- ings close to the shelter, another observer, etc.). Accordingly, the following definition is offered: A numerical series representing the variations of a climatological element is called "homogeneous" if the variations are caused by and only by variations of weather and climate. -20 FIG. 26. Variations of January temperature at Moscow in the years 1860-1870. (Deviations from the normal, F) Obviously, it must be the first step of the climatologist to examine whether or not the series is homogeneous, in order to avoid inferences which are strictly false. This examination en- counters a great difficulty. There are real climatic variations from year to year. The example of Figure 26 shows the variations of average January temperatures at Moscow in the years 1860 to 1870. We know that the observations are incontestable. If we did not know this fact, we should not be able to indicate whether the increase of temperature, e.g., from 1862 to 1863, was caused by nature or otherwise. Therefore we have to conclude that it COMPARISON OF OBSERVATIONAL SERIES 131 is not possible to discriminate between natural and artificial varia- tions of an element if the observations of only one place are available. In other words, it is generally impossible to decide whether or not a series of observations is absolutely homogeneous. On the other hand, it is clear that the variations of weather and climate are not restricted to a single place, with the exception of a cloud- burst over one spot, or kindred phenomena. This principle be- comes better exemplified, of course, if averages over a month, a season, etc., are considered. In longer periods, local extreme devia- tions are compensated and the trend of the variation is maintained over large regions. 1 First, temperature may be taken as an example. New York City and New Haven, Connecticut, are 80 miles apart. The January temperatures of the ten years reproduced in Table 35 TABLE 35. THE QUASI-CONSTANCY OF THE DIFFERENCES BETWEEN JANUARY TEMPERATURES AT Two PLACES, 80 MILES APART New York City (40.7N, 74.0W, 314 ft) (7) New Haven, Conn. (41.3N, 72.9W, 106 ft) (H) A 7 ~ Deviations Y A# = Deviations H (1) Year (2) (3) H , F (4) (Y -f/)F (5) AF, F (6) A#,F (7) 1911 34.5 32 1 + 2 4 +3 5 4-2 9 4-0 7 1912 23.2 21.2 2.0 7 8 80 T u.i +0 3 1913 . . 39.7 38 2 1 5 +8 7 4-9 2 1914 31.1 29 3 1 8 +0 1 4-0 1 4-0 1 1915 33.8 330 8 +2 8 4-3 8 1 u.i Q 1916 35.1 330 2 1 +4 1 +3 8 4-0 4 1917.. . 32.1 30 6 1 5 + 1 1 + 1 4 2 1918 21.3 19.8 1.5 -97 9 4 2 1919 34.9 33 8 1 i +3 9 -4-4 6 6 1920 23.8 21.4 2.4 -7.2 -7.8 4-07 Mean 31.0 29.2 1.7 24.2 25.6 2.2 24.7 25.2 2.1 4.89 5.08 =t0.43 show great variations. The temperature in New York varies by chance by the same amount as in New Haven, i.e., 18.4 F. The differences vary by 1.6 F. 1 On the other hand, it is very unlikely that the same kind of artificial influences occur synchronously at two different places. 132 METHODS IN CLIMATOLOGY The mean deviations (average variabilities) are (column 5, 6): AF = 4.9; Atf = 5.1 where F means New York and H means New Haven. The average variability of the differences is, however: A(F- H) = 0.4 Thus, the differences of the temperatures can be called quasi- constant, because their variations are the twelfth part of those of the actual temperatures. Similar conditions are clearly demon- strated in the example shown in Table 36A, 2 for precipitation, even TABLE 36A. THE QUASI-CONSTANCY OF THE RATIOS BETWEEN RAINFALL IN JULY AT TWO PLACES, 11 MILES APART Boston, U. S. Weather Bureau (42.3N, 71.1E, 124 ft) (B) Waltham, Cambridge Water Works (20 ft) (W) (1) Year (2) Boston (B) ' in. (3) Waltham (W} in. (4) I B/W (5)* AS (6)* AW (7)* 1911 4.65 4.83 0.96 +0.69 +0.73 -0.02 1912. .. 5.16 4.38 1.18 > +1.20 +0.28 +0.20 1913. . . .2.69 3.29 0.82 % -1.27 81 16 1914. . 2.64 2.46 1.07 -1.32 -1.64 +0.09 1915. . . 8.85 9.87 1.07 +4.89 + 5.77 -0.08 1916 5.67 5.31 0.95 + 1.71 + 1.21 +0.09 1917. . 1.10 0.93 1.18 -2.86 3.17 +0.20 1918... 2.64 2.99 0.88 -1.32 -1.11 -0.10 1919 4.63 5.21 0.89 +0.67 + 1.11 -0.09 1920... ....1.56 1.78 0.88 -2.40 -2.32 -0.10 .3.96 4.10 0.98 9.16 9.10 +0.58 2(-) 9.17 9.05 -0.55 v = db 1.83 1.82 0.11 v r (B t W) = db 45%f v r (B/W) - db 11% * Deviations from the means of the period 1911 to 1920. f v r (B t W) denotes the average relative variability of the amounts at B and W. if here for this element a much smaller distance between the two places has been chosen. If precipitation is concerned, it is not 'Temperature data are taken from H. Helm Clayton, World Weather Records (Smithsonian Miscellaneous Collections, vols. 79, 90); precipitation data from X. H. Goodnough, "Rainfall in New England," (Journal of the New England Water Works Association, Boston, June 1930.) COMPARISON OF OBSERVATIONAL SERIES 133 the differences that count, but the ratios between the amounts at the two places. The explanation will be given later on. Here, the ratios are much less variable than the rainfall itself, as is indicated by the mean deviations (v) at the bottom of columns 5, 6, 7. In reality, this behavior of the variability of the ratios is not yet decisive. In this example, it is a question of the amounts, which means that relative, not absolute, variabilities (v r ) have to be considered. (See III. 9.) Therefore, the absolute variabilities must be divided by the respective arithmetic means (ni). These relative variabilities (v r ) read : v r (B, W) = 45% and Vr(B/W) = 11% So it is clear that in the example the ratios between the amounts can be assumed as quasi-constant, in comparison with the actual rainfall amounts. The quasi-constancy of the ratio B/W can also be shown by another very common representation. In Table 36n, the amounts TABLE 36e. FLUCTUATIONS OF RAINFALL AMOUNTS AS PERCENTAGE OF THE AVERAGE VALUES AS ILLUSTRATIVE (AFTER L. W. POLLAK) OF THE QUASI-CONSTANCY OF B/W Year Percentage of average Boston ^ 1911-1920 IValtham Ratio B/W B A W units of second decimal place 1911 117 118 0.99 - 3 1912 130 107 1.21 19 1913 68 80 0.85* -17 1914 67 60 1.12 10 1915 223 241 0.93 - 9 1916 143 130 1.10 8 1917 28* 23* 122 20 1918 67 73 0.92 -10 1919 117 127 0.92 -10 1920 39 43 0.91 -11 \1.02 ifi 3.96 in. 4.10 in. 0.97 \ 11.7 of rain in B and W are given as percentage of average, 1911 to 1920. The amounts of rainfall at Boston vary from 28% to 223% or by 195%, and those for Waltham by 218%, while the ratios of B/W vary by 37% only. 134 METHODS IN CLIMATOLOGY These examples show that the variations of average weather particularly have identical tendencies over rather large regions. If the temperature drops at a place a it may drop too at the place b (not too far distant from a), and the same is true for rainfall, cloudiness, etc. In other words, there is no reason why variations of differences or ratios should be systematically influenced by the average weather events. XII. 2. CRITERIA OF RELATIVE HOMOGENEITY As previously stated, a series of numbers representing the variations of a climatic element is called homogeneous if the varia- tions are caused by and only by climatic influences. Now we have shown that these cannot give rise to systematic variations of " differ- ences" or " quotients." The two facts together yield an exact definition of relative homogeneity, as follows : A dimatological series is relatively homoge- geneous with respect to a synchronous series at another place if the differences (ratios} of the pairs of the homologous averages represent a series of random numbers which satisfies the law of errors. To determine whether or not two series are relatively homoge- neous is now a problem of statisticaKmathematics. I want to give two criteria, 3 those of Helmert and Abbe. XII. 2. a. Helmert' s Criterion The criterion of F. R. Helmert, well known in geodesy, is ap- plicable to the present problem and offers a first estimation. If two consecutive deviations of the series have the same sign, we speak of a sequence. The number of sequences (4- + or ) is called S. A change of sign of two consecutive elements is called change (H or h). The number of changes is called C. Now it may be assumed that the elements of a series are not systematically influenced but represent random numbers. Devia- tions are calculated. The consecutive signs yield 5 sequences and C changes. Then Helmert's criterion indicates that 5 - C = with a standard deviation db AW - i V. Conrad, " Homogenitatsbestimmung meteorologischer Beobachtungsreihen," Met. Zeit., 1925, p. 482. COMPARISON OF OBSERVATIONAL SERIES 135 where n equals the number of elements. In other words: in the absence of a systematic influence - Vn - 1 S (S - C) S + A/n - 1 In the example of Table 35, the number n = 10 therefore Vn 1=3 and - 3 ^ (S - C) s + 3 Counting sequences and changes in column 7, we get : S- C = 3 - 6 = - 3 Thus, Helmert's criterion shows that the series of differences may not be systematically influenced. This means that the two series of January temperatures for the years 1911 to 1920, one taken at New York, the other at New Haven, are relatively homogeneous. 4 XII. 2. b. Abbe's Criterion Another criterion, that of Carl Abbe, which originated from physical research, has been suggested by the author. 6 There may be a series of deviations from an arithmetical mean : From this series, two other series are derived : tn 1 \ A *] 1 \ *] ^ \ j-7 2 I .^ 2 T i-l 2) B = (d, - dtf + (d* - d*Y + (<*<- d i+l y (d _, - <*)' + (d n - dtf = L" (di - d i+ tf =-! Series B is considered as cyclic, so that the last term is formed as the difference of the last and the first deviation. By this proce- 4 Here a series of only ten elements is given, for the sake of brevity. Generally, the numbers of elements should be rather greater to permit application of probability criteria. See F. R. Helmert, Die Ausgleichsrechnung . . . (2nd ed., Leipzig, 1907), p. 333. V. Conrad, Met. Zcit. t 1925. 136 METHODS IN CLIMATOLOGY dure, the numbers of items of the two series A and B are equal. Series B reads: t'n =n tn t n B = y (d d - ") ^ = y (d ) ^ I y^ f d ' i } ^ - 2 T^ d' d t-i *-! t-i t-i Now 53 (^) 2 53 (^-i-i) 2 t-i ==i as identical values. Therefore or B = 24 2 53 ^fiJi-M = 24 F The most probable value of 7-^ | j | j r\ if the deviations are random numbers and n is a great number. Under these conditions one can imagine that it is always possible to combine any product db ;+i with another term of equal abso- lute magnitude and opposite sign. Consequently, the series F determines whether or not there is a systematic influence. If there is none, or in other words, if the deviations are random numbers, F = in so far as n is a large number. In concrete cases, F only approximates the value of zero with increasing n so that 2A l (Appendix II contains the values of 1/V# for n = 1 to 119, so that the limits can be easily determined.) This is Abbe's criterion, which is fulfilled if the series in question is not systematically influenced. When we have the problem which arises from the series of differences or quotients, respectively, the application of Abbe's criterion is most valuable because it considers the sequence of the signs as well as the quantity of the single deviations. 6 Tables 37 6 A third criterion regarding groups of equal signs is given by V. Conrad, "Die klimatologischen Elemente," in K&ppen-Geiger, Handbuch der Klimatologic, vol. IB, p. 114. COMPARISON OF OBSERVATIONAL SERIES 137 TABLE 37. ABBE'S CRITERION APPLIED TO A SERIES OF JANUARY TEMPERATURES AT NEW YORK (Y) AND NEW HAVEN (H) (See Table 35, Column 7) Year - A(F - \dt - 1911 +0.7 0.4 .49 .16 1912 +0.3 0.5 .09 .25 1913 -0.2 0.3 .04 .09 1914 +0.1 1.0 .01 1.00 1915 -0.9 1.3 .81 1.69 1916 +0.4 0.6 .16 .36 1917 -0.2 0.0 .04 .00 1918 -0.2 0.4 .04 .16 1919 -0.6 1.3 .36 1.69 1920 +0.7 0.0 .49 .00 2.53 B 5.40 TABLE 38. ABBE'S CRITERION APPLIED TO THE RAINFALL SERIES OF TABLE 36A* (1) Year (2) (3) \dt -d,+i| (4) (5) (di <f'+i) J 1911 -0.01 .22 .0001 .0484 1912 +.21 .39 .0441 .1521 1913 -.18 .28 .0324 .0784 1914 +.10 .17 .0100 .0289 1915 -.07 .05 .0049 .0025 1916 - .02 .23 .0004 .0529 1917 +.21 .30 .0441 .0900 1918 - .09 .01 .0081 .0001 1919 -.08 .01 .0064 .0001 1920 -.09 .08 .0081 .0064 A = .1586 B - .4598 * The series of deviations in column 2 of Table 38 differs slightly from column 7 of Table 36A. The result is valid for both series because that of Table 36A gives: 2A/B = 0.77 and 38 give examples of the application of Abbe's criterion. Here the series of differences (column 7 of Table 35) and the series of quotients (column 7 of Table 36A) are examined for relative homogeneity. From Table 37 may be calculated: 2A B 2 X 2.53 5.40 = 0.94 As n = 10, 1/Vn = db 0.32 (See Appendix II). Therefore 0.68 s r = 1-32 > 138 METHODS IN CLIMATOLOGY if no systematic influence exists. The value 2A/B lies indeed between these limits. Therefore it is probable that the two series are relatively homogeneous. The criteria of Abbe and Helmert arrive at the same result. From Table 38 we derive: 2A _ 2 X 0.1586 B " 0.4598 ~ U<b and because 0.68 S-Sl.32 > Abbe's criterion is fulfilled and there is no reason to assume a systematic influence upon the A(3/WO series. 7 Counting the sequences (5) and the changes (C) of column 2 (Table 38) it follows that: S = 3, C = 6, and 5 - C = - 3. As A/n 1=3, Helmert's criterion is fulfilled if - 3 S (S - C) s + 3 and again the two criteria agree with one another. If the relative homogeneity between the series of an element at two places a and b is given, and a third station c, suitably located, is available, a conclusion can be drawn regarding the absolute homogeneity of one of the series. If the pairs of series (a, 6), (b, c), and (c, a) are relatively homogeneous, it is rather probable that each of the three series is absolutely homogeneous. Thus a sort of climatological triangulation can be accomplished. The larger the number of places included, the greater is the likelihood that the conclusions regarding absolute homogeneity will be true. The chief trouble with all these investigations is the scarcity of stations. The meshes of the climatological networks are gen- erally too wide, so that the application of the higher and exact methods meets with difficulties. Nevertheless, the examination of a pair of records for relative homogeneity is fundamentally im- portant if the climate of a region is to be described accurately. Only by this examination can the natural climatic changes be separated from the unnatural by means of an exact analysis. One should never forget that a leaking gauge can counterfeit the finest periodicity. 7 It may be emphasized that Abbe's criterion, in the form discussed here, cannot be applied to smoothed series. See V. Conrad and O. Schreier, "Die Anwendung des Abbe'schen Kriteriums auf geophysikalische Beobachtungs Reihen," Gerland's Beitrdge zur Geophysik, XVII (1927), 372. COMPARISON OF OBSERVATIONAL SERIES 139 XI L 3. REDUCTION OF CLIMATOLOGICAL AVERAGES TO A CERTAIN PERIOD XII. 3 . a . Method of Differences Hitherto it has been assumed that the observations at the various places are operated synchronously. Then the comparable- ness is guaranteed as long as the series are mutually relatively homogeneous. TABLE 39. TEMPERATURE CONDITIONS OF DIFFERENT PERIODS AT ST. PAUL, MINNESOTA (45.0N, 93.0W, 837 FT.) INSANITARY (1) Year (2) January F (3) Year (4) January op (5) Col. 4 - Col. 2 1883 0.9 1898 22.8 + 21.9 1884 7.5 1899 13.8 6.3 1885 .... 4.3 1900 21.2 16.9 1886 ... 3.7 1901 16.3 12.6 1887 0.7 1902 18.7 18.0 1888 -1.1 1903 15.3 16.4 Mean 2.7 18.0 15.3 Series of different periods, however, are not comparable with one another. A short example is given in Table 39. 8 If one did not know that the two series (columns 2 and 4) originated in the same place, one might suppose that the series in column 4 repre- sented the temperatures at a place 8 latitude farther south. In addition, the July temperatures are higher in the period of cold winters, and not only are the actual temperatures very different in the different periods, but also the annual ranges. Another example (Table 40) gives the amounts of rainfall during different periods at Maiden Island in the South Pacific. TABLE 40. RAINFALL AT MALDEN ISLAND (4S, 155W) IN DIFFERENT PERIODS Period Average amount inches Years 1890 to 1899. Years 1911 to 1920. Januarys 1890 to 1899. Januarys 1911 to 1920. .16.30 .41.97 . 0.55 . 5.55 Data drawn from R. DeC. Ward and C. F. Brooks, "The Climates of North America," Kttppen-Geiger, Handbwh der Klimatologic, pt. J. (1936). 140 METHODS IN CLIMATOLOGY There, to the best of the author's knowledge, the greatest known contrasts occur. From these examples, it is clear that the changes in the climatic elements with time are so effective that a comparison between average values of temperature, rain, cloudiness, etc., at different places can lead to a reasonable conclusion only if identical periods are considered. In view of the scarcity of climatological data, however, the restriction of comparisons to a certain period would seriously impede climatographic work. The solution of this funda- mental problem is found in the quasi-constancy of differences (quotients) of synchronous averages of different climatic elements, which has been discussed earlier. We take two more examples from Table 35: January New York ( Y) New Haven (H) (Y + H)/2 ( Y - H) 1916 35.TF 33.0F 34.1 2.1F 1918 21.3 19.8 20.5 1.5 Diff 13.8 13.2 13.6 0.6 This means that the January temperature changed by 13.6 F from 1916 to 1918, but the difference between Y and H changed by only 0.6 F. This behavior has been called "quasi-constancy" of the differences. Another example from Table 36A, shows the following con- ditions: RAINFALL, IN INCHES Boston Waltham Ratio B/W 1915 July 8.85 9.87 0.90 1917 July 1.10 0.93 1.18 In other words, while the rainfall was 9 times greater in the wettest July (1915) than in the driest July (1917) the ratio of the rainfall in Boston over that in Waltham changed by only 24%. Thus the quasi-constancy of differences (quotients) is evident, both by means of a general geophysical conception and by prac- tical examples. A climatological station S 9 has been operated N years, where N is a relatively large number, perhaps greater than 25. A second station a has been operated n years. Let n < N and 9 The letter 5 (Secular Station) has, of course, another meaning from that in the previous paragraph dealing with Helmert's and Abbe's criteria. COMPARISON OF OBSERVATIONAL SERIES 141 the n years lie within the period of N years. The station S is called a "normal" or " secular" station. Let the n-year average temperature at the station a be i(a, n) and the same n-year average at the station S be i(S, n) Then the difference d is d = i(a, n} - i(S, n) If this difference is assumed as quasi-constant, the short period- average <(a, n) can be reduced to the normal period N for the whole of which only the station S has been operated. The reducing equation reads: l(a, N} = i(S, N) +d i(a y N) is the average temperature at the place a reduced to the normal period. The temperature series of the following table may serve as an example: It is assumed that the stations New York (F), and New Haven (H) have been operated simultaneously only in the follow- ing January months: y, F H, F January 1912 ...................... 23.2 21.2 1914 ...................... 31.1 29.2 1917 ...................... 32.1 30.6 1918 ...................... 21.3 19.8 1920 ...................... 23.8 21.4 Average of 5 years ................. 26.3 24.5 d = l(H 9 5) - i(Y,S) = - 1.8 F If New York is the normal station which has been operated during the entire 10-year period 1911 to 1920, we get the equation: ) = l(Y 9 N)+d and here l(H, 1911 to 1920) = i(Y, 1911 to 1920) - 1.8 F According to Table 35 : i(Y, 1911 to 1920) = 31.0F 142 METHODS IN CLIMATOLOGY Therefore the reduced value i(H, 1911 to 1920) = 29.2F In reality, the station at New Haven has also been operated over the entire 10-year period. The observed average is by chance identical with the calculated value. It may be remarked that the 5 -year average at New Haven is nearly 5 F lower than the reduced one. Therefore the 5 -year average cannot be compared with a 10-year average of another station. The reduction to equal periods is fundamental for all climato- logical and climatographical purposes. XII. 3. b. The Reduction of Precipitation Series: Method of Quotients For series representing rainfall, the procedure of reduction to a normal period is analogous, but ratios are used instead of differences. A normal (secular) station 5 has been operated over the normal period of N years. Another station a, located at a small distance from S, has been operated n years, so during these n years simul- taneous observations have been conducted at the two stations. The nomenclature is (partly repeated) : N = normal period of the Secular Station S n = short period during which the station a has been op- erated (n < N}. P(S, N) = Normal amount of rainfall at 5 during the nor- mal period N p(a, n) = average rainfall at a during the short period n P(S, n) = the same for the normal station p(a, N) = average annual rainfall at a, reduced to the normal period p(a, N} is the unknown value. Then under the assumption of quasi-constancy of the ratio q: P(S, n) P(S, N} Therefore p(a, N} = q X P(S, N) To take an example from Table 36A : Let Boston Weather Bureau (B) be the "normal" station, operated during the entire period COMPARISON OF OBSERVATIONAL SERIES 143 1911-1920. At the Cambridge Waterworks, Waltham, Mass. (W), precipitation might have been measured for only five years, with the following results at the two stations: July 1913 B 2.69 W 3.29 1914 2.64 2.46 1917 1.10 0.93 1918 2.64 2.99 1920 1.56 1.78 sum 10.63 11.45 P(W, 5 years) = 11.45 inches P(B, 5 years) = 10.63 inches and p(W, 5 years) = 11.45 = p(W, 10 years) P(B, 5 years) 10.63 P(B 9 10 years) According to Table 36, the 10-year average for Boston, Mass., is 3.96 inches; therefore: p(W,lQ years) = 1.08 X 3.96 inches = 4.28 inches. As is seen from Table 36, the true 10-year average for Waltham is 4.10, so that the difference observed calculated = 0.18 inches The error is 4.39%. It is a rather good agreement if one con- siders that the actual 5 -year average for Waltham was 2.29 inches, against a true mean of 4.10 inches. The reduction, reasonably applied, always yields reasonable results, while the comparison of different periods leads to false inferences. XII. 3. c. Physical Explanation of the Method of Quotients A short discussion must be added to explain why "ratios" instead of differences are used for the reduction of rainfall amounts. There are two stations, a and 6, with the average rainfall amounts p(a) and p(b). The assumption is made that p(b) > p(a). This may be expressed as follows: p(b} = p(a) + A(a). In a given year, a precipitation greater or smaller than normal may occur at a, that is, k X p(a), where k is a coefficient greater or smaller than 1. The question is, what amount is recorded at b. If the differences were invariable, p'(V) = k-p(a) + &p(a). 144 METHODS IN CLIMATOLOGY Then A(a) would be the constant difference between the amounts at the two places. This assumption does not appear likely. The average incre- ment A/>(a) is caused at the place &, for example, by the contour of the region or some other physical feature. It is, therefore, much more probable that p(a) and the increment &p(a) are increased according to the ratio k. We have therefore to expect at the place b the amount kp(a) + kbp(a). Numerous examples which prove this consideration could be presented; hence it can be taken as certain that the ratios of the rainfall amounts at places not too far apart are quasi-constant, in contradistinction to their differences. XII. 3. d. The Reduction of Other Elements to a Given Period From the foregoing, it is clear that averages of temperature and precipitation at different places are comparable with one an- other only if they are related to an identical period. This postu- late is obviously valid for each climatic element, be it primitive, derived, or combined. There remains the question whether differences or quotients have to be chosen. Only a few oft the most important elements are analyzed in this respect, namely: a) Number of days with precipitation b) Cloudiness 10 c) Duration of snow-cover u d) Number of days with fog As the differences in the case of these various climatological elements yield better results than ratios, or results equally good, the method of differences is preferred because it involves easier calculating. XII. 3. e. Limits of the Method of Reducing Climatological Series to a Normal Period The basic principle of the " reduction " to a normal period is the quasi-constancy of differences or quotients. The mathematical " V. Conrad, "Zum Studium der BewSlkung," Met. Zeit., 1927, pp. 87-91. 11 V. Conrad, M. Winkler, "Beitrag zur Kenntnis der Schneedeckenverhaltnisse in den dsterreichischen Alpentendern," Gerland's Beitrdge zur Gcophysik, XXXIV, 473-511. COMPARISON OF OBSERVATIONAL SERIES 145 meaning of the word " quasi-constancy " is that the variability of differences and quotients is much smaller than the variability of the elements themselves. From the example in Table 35 may be derived : v (diff .) = 0.43 v (temp. F) = db 4.89 v (temp. H) = 5.08 As was said earlier, v (diff.) is about a twelfth of the variability of temperature. Obviously, the average variability of the differences (quotients) increases with increasing distance between the two places. The same effect occurs if one of the two places is shifted beyond a climatic divide. In both cases, the variability of the differences becomes greater and greater; continuously in the first case, and eventually discontinuously in the second. Apparently, the quasi-constancy disappears when the vari- ability of the differences (quotients) equals the variability of the element itself; then a reduction to a normal period is no longer reasonable. That is the theoretical limit of the reduction, and is expressed by the equation : v (diff., quot.) = v (element) If v (diff. or quot.) is called v(R) and v (element) is called v(E) then generally this equation is valid : v(R) = k-v(E) Only if k < 1 is a reduction theoretically admissible. The question is, at what value of k should the practical limit of reduction be assumed? Such limits must be determined em- pirically. In the writer's experience, the reduction should be made only when k ^ This result is, of course, valid for all elements. The increase of v(R) has been represented by Hann by means of an empirical, analytic equation, which reads: V (R) = a + 6-AE + c-bh where a, b, c, are constants, AE the distance between two places, and A/& the difference of height. 146 METHODS IN CLIMATOLOGY This equation is mentioned here because it appears in many manuals, and also because it can be misleading. At the limit: AE - Aft = the differences are related to the element of the " normal station" itself. Then, however, v(R) = 0, This does not occur in the equation above, because the con- stant a ^ O. 12 In Chapter IV, 2, phenomena of decay are discussed and an example is given to show how the variation of v(R) for cloudiness can be represented by an analytical equation which exactly ful- fills the boundary conditions. Lack of space prevents a fuller discussion of the problem; but the above hints should suffice for further investigations. Reduction to a normal period has to be combined with an examination of the relative homogeneity, the equally important investigation into the nature of a climatological series. Both procedures require the computation of deviations. Finally, it may be remarked that, in the opinion of many authors, very long series are comparable with one another without being reduced to the same periocL Practically, one may be lucky enough to deal with series which happen to be comparable. It is perhaps more likely with non-reduced long series than with short ones. In any case, however, especially with long series, the examina- tion of relative homogeneity is unavoidable. XII. 3. f. Additional Note. The Length of a Normal Period It is frequently asked, how long a normal period should be. This depends upon three main conditions : 1) How many years of incontestable observations of the normal station or stations are available. 2) How long the whole period must be in order to include the periods of the series which have to be reduced. 3) How great the variability is of the element in question. Because of these conditions it is not possible to define a priori how many years make a normal period. 12 For more details see V. Conrad, "Zum Studium der Bewdlkung," Met. Zeit., 1927, p. 87. COMPARISON OF OBSERVATIONAL SERIES 147 Table 40 gives examples of the enormous variability of rainfall at Maiden Island (4S). In view of this fact, perhaps 50 years may not suffice to report, even approximately, the variations possible at this place. On the other hand, the variability of temperature on this tiny island in the equatorial belt of the Pacific Ocean, is nearly negligible. With regard to temperature, a normal period of 2 years would be sufficient. Some scientists are of the opinion that the normal period should include at least one "Bruckner-cycle," i.e., about 35 years; that is, a normal period should not be shorter than 35 years. The follow- ing considerations lead perhaps to the same result. 1*0 140 120 LOO 10 20 30 40 50 60 70 80 90 100 YEARS FIG. 27. Increase of range of the extremes of temperature with length of period Birkeland and Frogner 13 have shown how the range of the extremes of temperature and rainfall increases with an increasing period of observation. Their investigation indicates that this correlation (extreme range-length of period) can be computed from the Gaussian law of errors. An example is given for the range of the extremes of temperature in Fig. 27. The range of a period of 10 years is taken as the unit; this unit range is increased by 20% during a further 10-year period of observation. The increase from 40 to 100 years is but about 20% as a conse- quence of the asymptotical trend of the curve. The increment from 100 to ISO years is 3%, and 2% for the following 50 years. These theoretical considerations are valuable hints, when making a reasonable choice of the length of a normal period, if a long series 11 B. J. Birkeland, E. Frogner, "Die extreme Variabilit&t der Lufttemperatur," Met. Zeit., 1935, p. 349. 148 METHODS IN CLIMATOLOGY of observations is available at all. But it is more important to take that portion of the long series which is incontestably homoge- neous than to strive for a " normal period" as long as possible. The upper limit of the length of a normal period should be dictated rather by the condition that the periods of the other series be included than by the intention of extending the normal period uncritically to infinity. It is undesirable for obvious reasons that a normal period of 100 years be taken, if the longest of the other series do not cover more than perhaps 20 years. These are some of the ideas which should be considered when choosing a " normal period. " In any case, its length should ex- ceed 10 years, because otherwise the conclusions from statistical methods are no longer useful. From all points of view, the length of a normal period should be about 25 to 35 years. 14 XII. 3. g. Interpolation of Missing Observations The quasi-constancy of differences or quotients, offers the possibility of interpolating missing or false values. Curve parallels for the variations of the respective element at two and more sta- tions are often useful, but numerical differences, and quotients in particular, should be preferred* This is the more essential as these calculations must be made in order to investigate homo- geneity, etc. Systematic and random errors are revealed by these simple methods, and can often be corrected to a certain degree. Important corrections should be made if possible by means of at least two pairs of stations, each of which includes the dubious station. 15 Only a full comparison ot the synchronous climatological series used yields reliable results. XII. 3. h. Coherent and Incoherent Climatic Regions and Their Statistical Evidence The increase of the variability of differences (quotients) with increasing distance between the places compared was mentioned in connection with the method of reducing climatological series to an identical period. Although previously not emphasized, the 14 The International Meteorological Organization adopted the period 1901-1930 as a '* standard period" for climatological normals (Warsaw 1935). See the Meteorological Glossary (London, 1939), under "Normals." 16 For details, see V. Conrad, "Die klimatoiogischen Elemente," p. 119, chapter 7, The interpolation of missing observations. COMPARISON OF OBSERVATIONAL SERIES 149 increase of the variability of differences occurs more markedly if the difference in elevation between the places selected is increased. The increment of variability is a continuous process under other- wise equal conditions. This has been shown in a study of cloudi- ness. 16 As was said before, one may represent the increase of v(R), the variability of the differences (quotients) with an exponen- tial equation. The values calculated with this equation are in close agreement with observation. Hann's formula, even if not in accordance with actual conditions in the immediate surround- ings of the normal station, shows too that the differences increase continuously. Since this behavior holds for two climatological elements, one can assume that the same is true for other closely related elements. This problem can be inverted in so far as the kind of variation of v(R) can be taken as a climatological characteristic of a region. A region within which the variability of differences (quotients) of elements at two places increases continuously with increasing distance (difference in elevation) is called climatically coherent. Lines or strips at which the variability of differences (quotients) varies discontinuously are called climatic divides. Therefore re- gions on different sides of a climatic divide are called climatically incoherent. These concepts, derived from the variability of the differences (quotients) of elements at two places, are of major importance for recognizing the climatical structure of a region, and are funda- mental for conclusions regarding the geographical distribution of the climatic elements. Consequently, the short and long range forecaster is most concerned with these ideas. A forecast fully valid for climatically coherent regions fails if it is extended to an adjacent, but climatically incoherent, region. 17 These considerations, coupled with those of homogeneity, lead to the method of correlation, most important in so many climato- logical problems, which will be discussed in the following inter- mediate chapter. The behavior of the correlation factor may be decisive whether or not climatic regions are coherent. This idea offers a great field for future investigations. 16 V. Conrad, "Zum Studium der Bewolkung," Met. Zeit., 1927, pp. 87 ff. 17 See also I. I. Schell, "On the Use of Climatically Coherent Areas in Seasonal Foreshadowing, " BAMS, vol. 23 (1942), pp. 182-183. INTERMEDIATE CHAPTER (C) CORRELATION C. 1. LINEAR CORRELATION MATHEMATICALLY, the ' ' quasi-constancy " of, for example, differences between average temperatures at two places means that a certain relationship exists between the tem- perature series at the two places. The ratio between the variabil- ity of the differences and the variability of the element itself is the measure of the association of the two temperature series. This measure is specifically climatological and should be stated in terms of those measures offered by the statistical method of correlation. A generally valid measure of the degree of association of two series in the simplest case can be easily established. Two series may consist of the following elements : X n and * Fi, F 2 ; - F n The elements of these series have to be compared with one another in pairs, i.e., X\ with Y\\ X% with F 2 ; etc. Because of the inequality of the arithmetical means (Jt and F), the direct comparison is possible only by introducing deviations : Xi - = Xi and F< - ? = y< There is no doubt that the sum of the products Xiy* is a measure of the degree of association of the two series. If the number of the elements is great and there is no relation- ship between the two series, the products xty* are random numbers which may vary between certain definite limits. Then it must be possible to find for each product another of equal magnitude and opposite sign. (See the analogous conclusion with Abbe's criterion.) This means that the sum of all products is zero if no association f=n exists. Furthermore the value of #,-y< depends, obviously, on i=l the number n of the elements of the series. Therefore the sum of the products has to be divided by n, in order to free it of the 150 CORRELATIONS 151 influence of the arbitrary number of elements and we get There are two further difficulties in comparing the Xi series and the yi series: 1) The different elements may be expressed by different units. 2) The scattering of the deviations of the two series is different. If the foregoing expression is divided by the products of the standard deviations of the two series, it is reduced to equal scatter- ing and the different units are eliminated at the same time. According to what has been said previously, - and * Therefore the measure of the degree of association of the two series is: n fejc, 2 _ 2y7 2 \ n ' n or finally which appears often in the form The expression r xy is called correlation coefficient. Obviously r xv = + 1 if the functions /(#;) and/(y t ) are identical or differ by a constant factor. In other words, the two series run parallel to each other. 152 METHODS IN CLIMATOLOGY If the trends of /(#;) and/(y;) are in exact opposition: as it is clear from the equation for r xy . These are the boundary values that occur if the association of the series is perfect and can be expressed by means of a normal analytical equation. As stated before, r xy = if no association exists between the two series. On the other hand, it is clear that a correlation coefficient greater or less than zero can often result by chance, even if no association between the two series exists, when only a small number of observations is available. In this case, it is unlikely that 2>Xiyi will become zero ; consequently the correlation factor r differs from zero also for unrelated series. Even with a greater number of elements r will be falsified in a similar way. It can be said, however, that then the true correlation coeffi- cient r, which would result from an infinite number of elements of the two series, lies with equal probability within or without the limits r +/ and r /. The value/ is the probable error mentioned previously (III. 6). It is: f = 0.6745 1 A/n With an increasing number of elements the probable error decreases more and more, approaching zero. (See Table for 1/Vn in Appendix II.) The smaller / is, the more significant is the respective correla- tion coefficient. K. Pearson gives the rule of thumb that correla- tion coefficients should be assumed as real only if they are at least 6 times greater than the probable error. According to the author's experience, this ratio is very useful in avoiding false conclusions. On the other hand, the limit r xy ^ 6f is too high, according to other authors, for estimating the reality of correlation-factors. In this connection, the opinion in the Meteorological Glossary (London, 1939, p. 54) is quoted: Unless it is confirmed by physical reasoning or other independent evidence, a correlation coefficient should not be accepted as significant, unless it exceeds three times its probable error, in which case the odds in favor of significance are 20 to 1. If a number of trial correlations are made, the chance of obtaining a single large coefficient is obviously greatly increased, and such an isolated coefficient should not be ac- cepted, unless it is four or five times its probable error. CORRELATIONS 153 In the main, it is of greatest importance to calculate not only the correlation factor but also the probable error. Only then is it possible to estimate the significance of a correlation factor. Generally, correlation coefficients considerably smaller than 0.5 have to be estimated most critically. 1 On the other hand, cartographical representation of correlation factors are very instructive. If a great region is covered by one sign and another region by the opposite sign and the two regions are systematically separated by a transition zone of mixed signs, in this case small correlation coefficients can also lead to further successful investigations. If the correlation of a pair of series covering a great number of years shows a certain sign, the series should be divided into parts. For each of them, a correlation coefficient should be computed. If the signs of the single coefficients are identical, a certain sig- nificance even of small coefficients cannot be rejected without argument. C. 2. EXAMPLE OF CALCULATING A CORRELATION The average air pressures in millimeters of mercury (Xi) on the one hand, and the numbers of hours with bright sunshine (F) on the other, are known for 10 months of February at a place in the temperate latitudes. % and T are the arithmetical means, x> and yi the deviations. (Table 41.) The opinion is frequently expressed that high pressure corre- sponds to fair, sunny weather in temperate latitudes. This as- sumption is examined by means of the values X iy Yi in columns 2, 3 of Table 41. The correlation coefficient is below 1/2, and is not even double its probable error. Therefore we have to conclude that a series of 10 pairs of average values does not offer any reason for assuming an associa- tion between air pressure and synchronous hours of bright sun- shine, as far as averages (sums) and the region in question are concerned. The plus sign ( +) could indicate the tendency in the sense of the opinion mentioned; it would be significant only if it reappeared also with other equivalent pairs of series. 1 Further remarks on this problem will be found at the end of C. 4. For an excellent model for this procedure, see Ellsworth Huntington, Tree Growth and Climatic Interpreta- tions in Quarternary Climates (Washington: Carnegie Institution, 1925), p. 161 ff. Attention is called also to the interesting "Dot Charts" of the relation between annual growth of sequoias and rainfall. 154 METHODS IN CLIMATOLOGY TABLE 41. EXAMPLE OF CALCULATING A CORRELATION (1) (2) (3) (4) (5) (6) (7) (8) No. Xi Yi xi y\ Xiyi x* y<* 1 43.7 60 -3.0 -26 + 78.0 9.0 676 2 51.5 133 +4.8 +47 +225.6 23.0 2209 3 50.6 72 + 3.9 -14 - 54.6 15.2 196 4 47.9 82 + 1.2 4 - 4.8 1.4 16 5 44.6 119 -2.1 +33 69.3 4.4 1089 6. . . .. .47.2 52 +0.5 -34 - 17.0 0.3 1156 7 52.9 81 +6.2 5 - 31.0 38.4 25 8 41.0 54 -5.7 -32 + 182.4 32.5 1024 9 37.1 83 -9.6 + 28.8 ' 92.2 9 10 50.1 129 + 3.4 +43 + 146.2 11.6 1849 46.7 Y = 86.5 *0* +484 r _ V228 X 8249 0.6745 = 0.187 H Note: For calculating/ see Appendix II. * To avoid a misunderstanding, it should be expressly remarked that the correlation factor does not indicate anything about the causal association of two series of numbers representing the varia- tions of two geophysical phenomena. The indications of the correlation coefficients are only of formal nature. This fact cannot be sufficiently emphasized. Two series can show a high correlation coefficient, because of a systematic observational mistake, or because the association is physically well justified, as, for instance, for synchronous values of air pressure at two places a few miles apart. Therefore it is not sufficient merely to calculate the corre- lation factors. If they are known, the really scientific investiga- tion begins with the physical explanation of the correlations computed. In the example of Table 41, the explanation is obvious. The tendency toward the occurrence of precipitation is evidently smaller with high pressure than with low pressure. But above a plain, especially above a city, fog is much more frequent with high pressure conditions in winter. The final result is that a direct connection between air pressure and the number of hours with bright sunshine is lacking in this special case. CORRELATIONS 155 C. 3. SIMPLIFICATIONS IN COMPUTING CORRELATIONS The number of applications of the method of correlation is un- limited in climatological problems; so some simplifications of the technique of calculation may appropriately be indicated here. The computation of the deviations (xy) is sometimes weari- some. If the simple definitions previously mentioned are con- sidered, the following identities result: Xi = Xi - X and y t = Y* - Y E 1 ; = nX Therefore and and and T = n? i = n? E 3cy.- = E ^i^ - nX? (see footnote 2 ) L v 2 _ V P Xi 2^ V/ ^ Y^2 V Y 2 *t V2 \ i A^ Z^-^-i ^-^ L i i 1 n n n E y* 2 = E ( Yi F) 2 = E FV 2 wP 2 i i i - F) (Definition of x< and Then Because and and we get: i i The further transformations are analogous. nJPF 156 METHODS IN CLIMATOLOGY These expressions substituted in the original equation : n x<y< 1 yield X* - wl 2 ) F< 2 - This formula uses only the quantities of the original series, avoid- ing the calculation of deviations. If we look at the numbers in columns 2 and 3 of Table 41, the original items, a further trouble is apparent; the large numbers with which one has to calculate. This inconvenience also is easily avoidable, if similar, simple considerations are made as above. Each number of the two series is reduced by a number which equals the minimum of the respective series. That is 37.1 in the series of column 2, and 52 in the seYies of column 3. Thus we get small numbers for the calculation quickly and easily. TABLE 42. CALCULATING A CORRELATION BY MEANS OF THE ORIGINAL NUMBERS REDUCED BY A CONSTANT VALUE No. X/F/ YS* 1 6.6 8 52.8 43.6 64 2 14.4 81 1166.4 207.4 6561 3 13.5 20 270.0 182.2 400 4 10.8 30 324.0 116.6 900 5 7.5 67 502.5 56.3 4489 6 10.1 00 0.0 102.0 o 7 15.8 29 458.2 249.6 841 8 3.9 2 7.8 15.2 4 9 0.0 31 0.0 0.0 961 10 13.0 77 1001.0 169.0 5929 JP' 9.6 ?' = 34.5 VXt'Yi' =3782.7 VXi'* 1141.9 ZFi 7 ' = 20149 n - 10; n%'?' - 3312.0; nX<'* - 921.6; wlY' 11902.5 470.7 ' - nf'*) 1347.9 + 0.349 CORRELATIONS 157 The example of Table 42 is identical with that of Table 41. The correlation between pressure (Xi) and sunshine hours (F) is calculated with these original numbers instead of the deviations Xi and y^ Furthermore, Xi and F are reduced by the minimum values of the respective series. The new nomenclature is : Xi - 37.1 = X^ and F, - 52 = F/ The other details of Table 42 are understandable without ex- planation. The results of the calculations in Tables 41 and 42 are, practically, identical. C. 4. REGRESSION EQUATION The deviations from the arithmetical mean Xi and yi (see C. 1.) can be plotted in pairs on a graph paper representing a Cartesian system of coordinates with the origin x = and y = 0. The points Xiyi can be scattered in such a way that they cover the quadrants of the coordinates with the greater uniformity the greater the number n of the pairs of deviations is. This is not of interest here. On the other hand the dot-diagram can show a crowding of the points, indicating a linear proportionality in the simplest. (See Fig. 28.) Then we can try to determine a straight line y = bx which passes the origin and is fitted to the point-cloud as well as possible. If x should be an analytical linear function of y then y bx would be exactly zero. In this problem, however, this is not valid, and the quantity b must be determined so that it makes the differ- ence y bx as small as possible. 3 Since the signs of the deviations 1 The reader may profit by another very expedient method of explaining, kindly communicated to the author by L. W. Pollak, Dublin, Ireland, in a letter. I quote: "We have observed n pairs of phenomena Xi, Yi. Forming the means of Xi and Yi respectively, we obtain (i) n n The values X t Y define in the graphical representation of our Xi, Yi a point M which as the 'centre* of the dot diagram plays an important role. 14 Further, we compute the deviations of the Xi and Yi from their arithmetical means, obtaining Xi -Xi- Xi - Xi -X, yi - Yi -?i-Yi-Y (2) Plotting the Xi, yi means nothing but shifting the origin of the codrdinate system of the Xi, Yi dot diagram to M. "Now we intend to approximate our deviations #,, yi (in the codrdinate system with 158 METHODS IN CLIMATOLOGY do not play any role in this problem, only their absolute amounts, b should be chosen so that the sum of the squares of y bx is as small as possible. This is reached if 2(y bx) 2 becomes a mini- mum. The condition is fulfilled, if and 22 (- x)(y - bx) = or Sry = Therefore . n The equation y = bx contains the arbitrary assumption that x is the independent variable. One can also write : x = b'y and 6'=i M as origin) by a straight line y = bx + d (3) so that the sum of the squares of the distances of each dot from our straight line, meas- ured in the direction of the /s, is a minimum. "Therefore S[y< - (bxi + d)J = Min. (4) and accordingly dd or The second equation of (6) gives directly: nd and rf = as S#i = and 2yi = 0. (The sum of the deviations from the arithmetical mean, re- garding the signs, is zero). Our straight line required passes through the origin or the 'centre 1 Af." CORRELATIONS 159 If we consider that the standard errors n and the correlation coefficient TABLE 43. ANNUAL VARIATIONS (DEVIATIONS) OF THE LEVEL OF VICTORIA NYANZA (0N, 32E), (xi inches) and of the Annual Sunspot Numbers (y) (After Sir Napier Shaw) (1) (2) (3) (4) (5) (6) Year x\ yi Xt* yS xtyi 1902 -18 -35 324 1225 630 1903 5 -16 25 256 -80 1904 10 2 100 4 20 1905 7 23 49 529 161 1906 21 14 441 196 294 1907 13 22 169 484 286 1908 2 9 4 81 18 1909 4 16 1910 - 7 -21 49 441 147 1911 -15 -34 225 1156 510 1912 -19 -36 361 1296 684 1913 -11 -39 121 1521 429 1914 -10 -30 100 900 300 1915 - 4 7 16 49 -28 1916 7 17 49 289 119 1917 27 64 729 4096 1728 1918 19 41 361 1681 779 1919 24 576 1920 - 5 - 2 25 4 10 1921 -13 -15 169 225 195 2 3317 15025 6202 mean 166 751 310 r xv = - ^ = + 0.88 <r, = Vl66 = 12.9 b xy = r- = 0.414 x = 0.414? <r y = VlSl 27.4 cr v 6 tf x = r = 1.869 y = 1.869* 160 METHODS IN CLIMATOLOGY then T 0*V J 7/ ^JC = r and Z>' = r <r x or y Therefore we get : (Ty - 0"x y r x and # = r y G x Gy These equations have been called regression equations by Sir Francis Galton (1886); 6(6') is called the regression coefficient. An example (Table 43) 4 serves as an illustration. Let %i denote the deviations (inches) from the average level of Lake Victoria (Victoria Nyanza) in Africa, and let y denote the deviations from the mean of the annual sunspot numbers (period 1902-1920). The values of columns 2 and 3 in Table 43, # and y,- f are plotted on graph paper. The values # mean the variations of the lake level (horizontal axis) and the y the variations of the sun- spot numbers (vertical axis). This dot diagram (Fig. 28) shows clearly that there is a rather strong correlation between the two phenomena. The two regression equations mentioned above are represented by two different straight lines irf Fig. 28. The smaller the angle between the two regression lines, the higher the degree of associa- tion and the greater the value of the correlation-coefficient r. If r = 1, the two regression lines coincide. If r = the one line is identical with the X-axis and the other with the F-axis (y = ; x = 0). As seen from Table 43, the correlation coefficient is high, and from Figure 28 it is clear that the angle between the regression lines is small in consequence of the strong association. Such cases are very rare in the realm of meteorology and climatology. The method of the curve-parallels is often misleading, and dot-charts are by far preferable to this method. Calculating correlation- coefficients and probable errors is never wasted time if the dot- chart is somewhat encouraging. The correlation coefficient, however, and the probable error, do not yet entirely satisfy the climatologist (as we have mentioned before). The critical estimation of the true value of a correlation coefficient is difficult. W. H. Dines says: 5 4 Taken from C. E. P. Brooks, Climate through the Ages (New York, 1926), p. 415 ff, and Sir Napier Shaw, Manual of Meteorology, I, 284, Fig. 116 and table on the same page. The continuation of the example concerning the partial correlation is also taken from C. E. P. Brooks' discussion of the problem. 6 The Computer's Handbook (Met. Office, London, 1915), p. V 39. CORRELATIONS 161 No precise classification of correlation coefficients can be made. As- suming that they depend on as many as 50 independent cases, one may say roughly that values under 0,30 are hardly significant, values between 0.30 and .70, prove a moderate connection; values between .70 and .90, a close connection, and values over .90, a very intimate connection indeed. It is also apparent that, if the coefficient is very high, fewer observa- tions are necessary to establish it, since 1 r 2 is then small. 6 -20 _i , LAKE \B 3) LEVEL ZQ 40 60 FIG. 28. Dot-chart and regression lines of the deviations from mean annual level of Victoria Nyanza [0 (Eq.), 32E] and from the mean number of sun- spots. The mean level of the lake and the mean number of spots are taken as axes. (After Sir Napier Shaw) From this classification, the coefficient of the example in Table 43 is only " close, " and C. E. P. Brooks expresses this fact by saying that " there is a good, but by no means perfect, agreement between the variations of the level of Victoria Nyanza and sun- spot numbers/' We can assume that " variations of level are 'caused' partly by variations of sunspots and partly by other factors, which for the sake of illustration we shall suppose to be independent of sunspots." This portion of the variations of level is, however, 1 - r 2 = 1 - (0.88) 2 = 0.23, or 23%. C. 5. PARTIAL CORRELATION In the foregoing, the problem of two variants which show a certain degree of association has been discussed. This case, 6 See probable error. 162 METHODS IN CLIMATOLOGY apparently, is the most important for the climatologist. The general problem, however, is that there are m variants, mutually dependent upon one another. The question is, how and with what accuracy can x be represented by the other variants, y, z, The method starts from the aforementioned correlation coeffi- cients and analyzes the complex influence of each of the series of variables upon x, so that the partial influence of the single variables upon x is separated one from the other. Therefore we speak of " partial correlation/' If, for instance, three variants are associated one with another, the problem is to find how the second variable is correlated with the first, when the third is constant, and so forth. 7 Now we return to the example of Table 43, which indicates that the variations of the level of Victoria Nyanza (x) are closely associated with the synchronous variations of the numbers of sunspots (y). In this example, especially, the need of a connecting link in the form of a third variant is evident: this is the rainfall. Its varia- tions (z) are known, and the correlation between z and x is indeed high : r x . = -.92 The coefficient between rainfall variations (z) and the variations of sunspot numbers (y) is: r yz = + 0.80 Theoretically two alternatives exist : 1) The sunspots influence the rainfall, and its variations cause the variations of the level of the lake. 2) There is a direct connection between sunspots and lake level. One should discriminate between these two alternatives. For this purpose, the partial correlation coefficients are calculated. We repeat the nomenclature : if T X y = the coefficient between lake level (x) and sunspots (y) r xg = the coefficient between lake level (x) and rainfall (z) r yz = the coefficient between rainfall (z) and sunspots (y) 7 C. U. Yule, An Introduction to the Theory of Statistics (2nd ed., London, 1912) and F. M. Exner, Ueber die Korrelations Methode (Jena, 1913). CORRELATIONS 163 the partial correlation coefficient 8 is given by the equation ' XV """" 1X2 ' ' UZ r xy.z From the example we get : 0.88-0.92-0.80 '*"" = 0.39-0.60 = + ' 61 The meaning of r xy . z is that " the effect exerted through rainfall " is "eliminated." In other words, the correlation between x and y is examined at constant rainfall. The relatively small correlation coefficient r xy . 2 shows that the major part of the effect on the level of the lake is caused by rainfall ; that corresponds well to common sense. On the other hand, " there is still an appreciable effect due to some cause independent of rainfall, most probably rate of evaporation." The expression for r xv . z has been given. The two other coeffi- cients are easily derived from this equation. We exchange y for z, z for y, and get from and an exchange of x for y, which in turn yields, fyz.x from r xe . y 8 D. Brunt, The Combination of Observations, (Cambridge, England, 1931), p. 175, gives an interpretation of the partial correlation coefficient which may be quoted here as a very valuable addition to the above-mentioned explanation. For the convenience of the reader, Brunt's symbols are replaced by those used here in the text. I quote: "We further require to know the correlation between x and y when each has been corrected for the correlation with z. This is written r xl/ .,, indicating that the effect of the variable 2 is eliminated. Since the regression equations of x and y on z are x r x , 2 = 0, y r y , 2 = <r g <r f fxy.i is the coefficient of correlation between x r xv ~ 2 and y r v , ~ z." 164 METHODS IN CLIMATOLOGY Thus we get three partial correlation coefficients : T xy ?xz ' fyz ?xz.y fyz.X == The respective standard deviations are : (fx.yz == 0W-1 f xy'\- The following six regression coefficients correspond to the three partial correlation coefficients. t + h Y (Tx ' vz h - r . (Tx ' vz Uxy.z ' xy.z ^xz.y ' xz.y Gy.xz &z.xy h - r (7v ' xz h - r (Ty ' xz u yx.z 'xy.z u yz.x fyz.x Gx.yz Gz.xy h v Vz.xy , _ Vz. Uzx.y 'xz.y Uzy.x 'yz.x* &x.yz ffy. Finally, the regression equations read : z = b xz .-x The example mentioned above shows that the method of the partial correlation is well suited for discriminating between the individual influences of the different variables. Therefore the method of partial correlation is much more than a kind of repre- sentation: it can be a successful method of investigation. For CORRELATIONS 165 want of space, these remarks about linear correlations must suffice. Generally, the common climatological problems can be solved by the methods indicated. L. W. Pollak's application of ' ( autocorrelation " 9 is of great importance for determining hidden periodicities as well as the most significant problems of long-range forecasts as far as higher statistics are concerned. This method correlates two portions of the series with the series itself , which may briefly explain term and aim of the method. 9 L. W. Pollak, "On Cycles of Pressure Especially in the Neighborhood of Symmetry Points," QJRMS, vol. LXVI, no. 287 (1940). CHAPTER XIII GRAPHIC COMPARISON OF CLIMATOLOGICAL ELEMENTS. ISOGRAM. MAP OF ISOLINES THE PROBLEM is to represent three variables on a two-dimen- sional plane. A well-known solution is the representation of contour lines on a map. The first variable is the geo- graphical longitude, plotted on the abscissa. The second variable, the latitude on the ordinate. Points of equal elevation (the third variable) are connected with one another by the contour lines. This analogy leads to the understanding of the method of repre- 60 -if FIG. 29. Isopleths representing the annual course of air temperature in the high northern latitudes. (After H. Mohn and W. Meinardus) sentation. An example of the application to climatological purposes is given in Fig. 29. The months are plotted on the X-axis, and the geographical latitudes on the F-axis. The monthly tem- peratures of the parallels may be known. For 85N, e.g., the 166 COMPARISON OF CLIMATOLOGICAL ELEMENTS 167 numbers read : Jan. Feb. Mar. Apr. May Jn. Jl. Aug. Sept. Oct. Nov. Dec. C -37 -37 -32J -25 -12 -2 -2 -11 -22 -29 -34 These temperatures are plotted on the map at the intersections of the verticals (indicating the mid-points of the months) and the parallels. Now the points of equal temperature are connected by curves as in Fig. 29. Along these lines, the temperature is con- stant. They are called isotherms for temperature ; (isohyets for pre- cipitation, isonephs for cloudiness, isohels for sunshine, etc.) The specific representation of three variants of which, e.g., the first is the characteristic of an element, the second the time, and the third a linear dimension is called isopleth-diagram or iso gram. Figure 29 takes the place of a table of 7 rows and 12 columns; besides, 1) by interpolation the annual course of temperature can be taken from this representation for any latitude be- tween 60N and 90N. 2) The vertical lines mean the middle of the months. Thus, by interpolation, the temperature-decrease from 60N to the pole can be read off along the vertical for any date of the year. 1 On the other hand, one should bear in mind that the accuracy of interpolation is very poor. Be that as it may, the isopleth- diagrams are often valuable illustrations of climatological condi- tions. The most important and indispensable application of this general method is the map of isolines. Isolines are fully analogous to contour lines, or to the equipotential lines used in physics. From this analogy, laws and rules for drawing isolines can be easily deduced. Closed isolines which fully surround a region indicate that this is either depressed or elevated. Isolines of equal value, running parallel, border an elevated ridge or a de- pressed furrow, etc. These principles, translated from the 1 Attention should be drawn to a special kind of isograms, which show the daily course as well as the annual. Irving F. Hand, in "A Summary of Total Solar and Sky Radiation Measurements in the United States," MWR, vol. 69 (April 1941), pp. 95-125, publishes a great collection of this kind of isograms, which present the best pattern for such graphs. They are extremely clear and legible, because they are not overburdened. The hours of the day are plotted on the J\T-axis, the months on the F-axis. L. W. Pollak gave an original application of the method of isograms, constructing iso-stereograms which are certainly of great general value and are also useful for didactic purposes (L. W. Pollak, Plastische Tabellen [Plastic Tables], Prague, 1914). 168 METHODS IN CLIMATOLOGY language of the contour-lines into that of climatology, mean for instance: high and low pressure region, ridge of high pressure, trough of low pressure, saddle between two high pressure regions, and similar characteristics for each element which is represented by a map of isolines. Isolines representing different values cannot cross one another. If the trend of an isoline is followed in a given sense, clockwise or counter-clockwise, decreasing and increasing intensities (amounts) always remain on the same side. Every map of isolines should be strictly examined, according to this principle. 2 Isolines should be drawn so that the difference of the intensity (amount) is constant between any pair of adjacent lines of the same map. Isolines which deviate from this general rule should be especially marked (e.g. by broken lines, if the regular are full). The mutual distance of two isolines is inversely porportional to the degree of variation (gradient) of the element. The " den- sity " of the isolines is proportional to the intensity of the spatial variation of the element. The linear interpolation of the intensity (amount) of an ele- ment between two points (places), is a necessary expedient in default of a better one. But the climatologist should realize how rough this approximation is. This ^circumstance on the one hand, and the mass of errors with which the average values of the ele- ments are infected on the other, should warn everybody against an exaggerated accuracy in planning and drawing such maps. It must also be emphasized that climatological networks usually have wide meshes. Thus, many lines are drawn over a distance of hun- dreds of miles, with only two or three points of known values. It is regrettable that, sometimes, neither is the homogeneity examined nor a reduction to an equal period made. These omissions have not deterred students from drawing isotherms for every F, in spite of the fact that the actual exactness of these observations is about 2 to 3 F. Consequently, the difference of intensity (amount) between two consecutive isolines should be weighed according to the accuracy of the data at hand, and to the openness of the network of stations. The map of isolines must never be overburdened. Therefore the use of interpolated isolines between those based on observa- tions in order to give a better appearance to the map is strictly ruled out. Such inserted parallel lines are nothing but an un- *See also the short instruction in A. F. Spilhaus and J. E. Miller, Workbook in Meteorology (New York, 1942), p. 1. COMPARISON OF CLIMATOLOGICAL ELEMENTS 169 necessary and arbitrary interpolation. Finally, the trend of the isolines should be chosen as simply as observations permit. Fanci- fully embellished trends mislead the layman, and are little ap- preciated by the expert. As far as the projection of maps is concerned, Sir Napier Shaw and V. Bjerknes arrive at the same conclusions: Shaw recommends in his Manual (vol. I, p. 262): ". . . three conformal projections, two polar planes extending to 67^, an adjusted conical projection for the region between latitude 67^ and 22^, and for the equa- torial regions between 22JN and 22|S, projection on a cylinder adjusted to be conformal." The number of applications of the method of isopleths and isograms can be increased arbitrarily, like the number of derived elements. Therefore, in the following section and in Part IV, only a few hints can be given for making maps of isolines and some difficulties mentioned. CHAPTER XIV METHODS OF ANOMALIES XIV. 1. TEMPERATURE DISTRIBUTIONS IN MOUNTAINOUS REGIONS TEMPERATURE varies with latitude according, to radiation conditions and the length of the day; it varies also with the distribution of land and water. Moreover, it varies with altitude about one thousand times as rapidly as with latitude. That is the difficulty. It is fortunate in this respect that about three fourths of the surface of the continents does not exceed 1300 feet, and that 58% is below 660 feet. In relation to these numbers, the mountain ranges appear only as disturbances of the continental surfaces, as far as the whole surface of the earth is concerned, or at least great portions of it. Therefore, the variations of tempera- ture with altitude have little influence on the trend of the iso- therms, on the whole. t To describe the climate of a rrtountainous country is another matter; for here the differences in altitude and relief play a decisive role, and the climate is determined by these factors. XIV. 2. EXAMINATION OF LAPSE RATE The variation of temperature per unit of height is called lapse rate. It is positive if the temperature decreases with height and is negative when it increases. The latter phenomenon is usually called temperature inversion. Two methods of calculating the lapse rate are used : 1) The temperature difference of two places at different altitudes is computed, and divided by the difference of height. The single lapse rates are averaged. 2) As described in an earlier chapter, temperatures of the different stations are arranged according to their respec- tive altitudes, then the elevations and the corresponding temperatures are averaged within certain intervals of altitude. (See IV, 4. b.) Thus a temperature-altitude curve is obtained, from which average lapse rates can be 170 METHODS OF ANOMALIES 171 taken for each interval, as well as for the entire difference of height. This latter method seems to be better and more accurate. The advantage of the first method is that a very small number of pairs of stations yield a first approximation. 1 XIV. 3. MAPPING TEMPERATURE CONDITIONS OF A MOUN- TAINOUS COUNTRY (REDUCTION TO A GIVEN LEVEL) The most common means of representation is by reduction to a given level. J. Hann suggested reducing temperatures in a uniform manner, independent of season and region, under the assumption of a lapse rate of 0.5 C per 100 meters or 3 F per 1100 feet. This suggestion should be generally accepted. Although reduced iso- therms are not satisfactory in a country with great "relief energy" 2 and great differences in elevation, at least the comparableness of such isotherm maps would result. It is regrettable that uniformity is still lacking. Following Hann's rule, the temperature reduced to sea level, / , is computed by the formula t = (t + 0.005 A)C if / is the average temperature in C at an altitude of h meters; or to = (t + 0.002743-A)F = (/ + 3/1100-A)F if h is expressed in feet. Frequently, sea level is not the most advantageous plane, but any other level, e.g. the average level of the valley bottom, or that of the plain or highland adjacent to the mountains, may often yield more plausible temperatures. The reduction of temperature when there are great differences in altitude is no longer a correction but a method of projection onto a plane. This projection should offer: 1) A survey of the geographical distribution of the tempera- ture. 2) The possibility of interpolating the temperature at a place of known altitude from which no observations are available. 1 For numerical comparison between the two methods, see J. Maurer, Das Klima der Schweiz (Frauenfeld, 1909), p. 154. 1 "Relief energy " is the sum of the differences in height along a cross section through the region, divided by the length of the cross sections. Obviously, this ratio is zero for a plain. Many authors relate the "relief energy" to a certain area and not to a length. (H. Wagner, W. Meinardus, Mathemat. Geogr. 1938, p. 349). 172 METHODS IN CLIMATOLOGY At first, this map may be disappointing because it contains fictitious temperatures which give a false picture. Only a con- version of the temperatures by means of the altitudes (which are mostly lacking on the maps) gives the actual temperatures. If the picture presented by the reduced isotherms is unsatis- factory, what of the possibility of interpolation? Here, too, the method does not give good results for winter in temperate latitudes. The reason is simple, since temperature inversions are more or less common above the valley bottom. This means that the average lapse rate is negative, and the average temperatures increase with height. It is cold in the valley bottom and much warmer at higher levels. The consequences of the reduction of temperature to a certain level under these conditions are illustrated by the following example: The temperature at a place A at the bottom of a valley at 440 m reduced to sea level is 4.2C. The actual temperature of a place 5, 746 m high, located on the slope, should be interpolated from the map. Then / = t - 7.46 X 0.5 C = - 4.2 - 3.7 = - 7.9C In reality, observations at *are at hand and the true January temperature is 3.7C, which is about 4 C higher than that interpolated from the isotherm map. The explanation is simple. We have calculated with a uniform lapse rate of 0.5 C/100 m, whereas in reality there is an increase of temperature, which can- not be considered in the procedure of a uniform interpolation. Consequently, one must conclude that the reduced isotherms do not yield a suitable representation of the temperature distribu- tion in a mountainous region. This conclusion, it should be emphasized, applies only in the case of a mountainous region. In open land, also in hilly regions, where small differences in height are more or less negligible in com- parison with the respective horizontal distance, the reduction to a fixed level may be accomplished with good results. XIV. 4. ACTUAL TEMPERATURES Owing to the great disadvantages of the method of reduced temperatures, many authors are led to give ''actual temperatures," i.e., the true average temperatures as observed at the altitudes of the respective places. METHODS OF ANOMALIES 173 Two alternatives exist : 1) There is a huge country with a high mountain range which is relatively not extensive in area, and the number of mountain stations is small (United States). 2) There is a country the greatest portion of which is covered with high mountains (Switzerland). In problem 1, the actual temperatures at the elevated stations do not affect the general picture of isolines much, since the moun- tain stations are few. In the open, hilly regions, the representa- tion of the temperature distribution differs but little from the reduced isotherms, because of the small differences in height. In problem 2, the actual isotherms could be drawn only if the density of the network were exceedingly great. Otherwise the curves would cross crests and abysses, valleys and ridges, regard- less of the variations of temperature. Yet just these variations should be represented. A clearly legible number, giving the actual average temperature, beside each place-name (similar to the method of indicating the altitude on orographic maps), would be a much better expedient than lines, the trend of which is by no means certain and cannot be guaranteed, since the number of stations is never sufficient, owing to the great " relief energy/' It is generally recommended that numbers^characterizing the represented elements be added beside the station names (or station symbols) on maps of isolines, in any case. Only then is the reader able to control the trend of the lines and to change it according to his individual point of view. XIV. 5. STANDARD CURVES. ANOMALIES. ISANOMALS As already discussed, the influence of altitude upon the geo- graphical distribution of temperature should be eliminated by a sort of projection called reduction to a fixed level. Another attempt in this direction, utilizing the actual temperatures, has been made. It has been shown in the foregoing that neither the one nor the other method yields satisfactory results. The discussion in Chapter IV. 4, about "curve fitting/' con- cluded that the graphical Cartesian representation of two variables, an independent (e.g., altitude) and a dependent (e.g., temperature), is important for climatological purposes. 174 METHODS IN CLIMATOLOGY These dot charts are interpolated either graphically or by means of analytical equations. Both methods permit the determi- nation of the values of the dependent variant for every value of the independent. Thus convenient tables can be constructed from which the values of the dependent variant can be taken for equi- distant values of the independent variant. A clipping from a table representing the average temperature-height curve above the bottom of a mountain valley was given in Table 16, as well as the corresponding graph Fig. 14, which can be called the standard curve. The numbers in the table represent the standard distribu- tion of the element with respect to the region in question. On the other hand, the actual value of the element at the place is known, as also its elevation. The difference, viz., actual value of the element minus standard value, is called anomaly. These anomalies are independent of altitude in the present case, because this is fully considered by means of the standard curve. Lines connecting places of equal anomaly are called isanomals. XIV. 6. APPLICATIONS OF THE METHOD OF ANOMALIES AND ISANOMALS * XIV. 6. a. Vegetative Period and Altitude The length of the vegetative period in Switzerland may serve as the first example. 3 As was said previously, the duration (number of days) of a temperature above 43 F, derived from an average annual course of temperature, is called the vegetative period. These periods have been calculated by a simple interpolation formula (see VI. 2) for 168 stations in Switzerland. Their altitudes vary between about 800 and 8200 feet. The entire country, with mountains up to nearly 16,000 feet, has an east-west extension of about 186 miles, and a north-south extension of about 124 miles. This is, indeed, a mountainous country divided into a northern and a southern part by the crest-line of the Central Alps. From all the 168 stations together, suitably grouped according to elevation, the following analytical equation for the relation be- tween the duration in days (d) of the vegetative period and the 3 V. Conrad, " Isanomalen der Andauer einer bestimmt vorgegebenen Tempera tur," Geografiska Annaler, 1929, p. 299. J. Maurer (Pas Klima der Schwciz, Frauenfeld, 1909, p. 68 ff.) gave anomalies and isanomals of annual average temperatures in Switzerland. METHODS OF ANOMALIES 175 altitude in meters (ft) is obtained : d = 268 - 0.07 h From this standard equation, the standard values are derived for altitudes for every 200 meters (Table 44). TABLE 44. STANDARD VALUES (DAYS) OF THE DURATION OF THE VEGETATIVE PERIOD IN SWITZERLAND Height (meters) Duration (days) Height (meters) Duration (days) 200 254 1200 .. . . 184 400 240 1400 170 600 226 1600 156 800 212 1800 142 1000 198 2000 128 The vegetative periods of two places, one north, the other south of the Alps, are compared in Table 45. TABLE 45. ANOMALIES OF THE DURATION OF THE VEGETATIVE PERIOD AT Two PLACES IN SWITZERLAND: Z NORTH OF THE ALPS; M SOUTH OF THE ALPS Height m Duration in days obs. standard Anomaly obs. standard days Z 355 221 269 243 243 -22 +26 M . . 355 The altitudes are by chance identical, so the two places have the same standard duration of 243 days. The actual duration has a deficit of minus 22 days north of the Alps; and a surplus of + 26 days south of the Alps. The numbers (Table 45) are really illus- trative of the climate in these two zones. Everybody who crosses the Alps, especially in the late winter, is surprised by the climatic difference between north and south. He leaves the northern region with its ground everywhere covered with snow; but when he has passed through the Gotthard tunnel for instance, he arrives at the famous Lake of Lugano, where the trees are in full blossom. The anomalies mentioned above give an exact picture of a sharp climatic divide between the places Z, north of the crest, and M, south of the crest at 335 m (1160 feet). A place, Ri f 400 feet higher than M, has +25 days; and Br, 1400 ft higher, has an anomaly of +30 days. The three places have similar exposure towards the south, and are well protected 176 METHODS IN CLIMATOLOGY against invasions of polar air. The differences are surprisingly small, in spite of the rather great differences in altitude. The examples show that apparently the influence of altitude upon the anomalies is practically eliminated. So it is possible to draw isa- nomals even in a region of great " relief -energy. 1 ' The anomalies combined with a standard element-height curve enable one to calcu- late the actual values for places where observations are available ; and, secondly, to interpolate with relatively small errors anomalies and actual values for any point. It is beyond the compass of this book to discuss the interesting picture resulting from application of the method of anomalies and isonomals. One example is shown in Figure 30, however: the isanomals of the duration of the vegetative period in Switzerland. In spite of the great differences in altitude, FIG. 30. Isanomals of the duration of the vegetative period in Switzerland. (After V. Conrad) on the southern slopes, a large region of the Alps is characterized by positive signs (a vegetative period longer than the standard duration) ; and north of the Alps, a zone of negative signs is seen. Not altitude but climatic exposure is represented. There is no doubt that the method of isanomals is, up to the present, the only method which eliminates the influence of altitude to a high degree, and is, thus, the unique method of graphically representing climatic conditions in a really mountainous region. METHODS OF ANOMALIES 177 It goes without saying that this method is practicable for any variable climatic element. XIV. 6. b. Temperature and Geographical Latitude The best known standard equation which correlates the de- pendent variable, the temperature (/) with the independent vari- TABLE 46. STANDARD DISTRIBUTION OF TEMPERATURE WITH GEOGRAPHICAL LATITUDE (After W. Meinardus and Hann-Stiring [5th Ed.]; Converted into F) * January April July October Year N90 -42 -18 30 -11 -7.6 85 -36.6 -15.7 32.5 - 8.0 -6.2 80 -26.0 - 8.9 35.6 - 2.4 1.0 75 -20.2 - 2.2 38.1 6.8 5.5 70 -15.3 6.8 45.1 15.3 12.7 65 - 9.4 18.9 54.3 24.6 21.6 60 3.0 27.0 57.4 32.5 30.0 55 12.4 35.2 60.3 37.2 36.1 50 19.2 41.4 64.6 44.4 42.4 45 28.9 50.7 69.6 52.7 49.6 40 41.0 55.6 75.2 60.3 57.4 35 49.3 62.6 78.4 66.0 63.0 30 58.1 68.2 81.1 71.2 68.5 25 65.7 73.8 82.2 76.3 74.5 20 71.2 77.4 82.4 79.5 77.5 15 75.2 80.1 82.2 80.6 79.3 10 78.4 81.0 80.4 80.4 80.1 5 79.3 80.2 79.2 79.3 79.5 Eq. S 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 79.5 79.5 79.3 78.6 77.7 75.7 71.4 65.7 60.1 54.1 46.6 41.0 35.8 30.7 25.7 19.8 12.6 8.6 7.7 79.9 79'7 78.6 77.4 75.2 71.2 65.7 59.4 54.5 46.4 43.3 35.2 27.5 19.0 7.5 - 6.7 -19.8 -28.7 -33 78.1 76.8 75.0 72.1 68.0 63.5 58.5 53.2 48.2 43.2 37.9 27.7 15.6 3.0 - 9.4 -23.4 -39.1 -49.9 -54 79.7 78.8 78.3 75.9 73.0 69.1 64.4 59.5 53.1 46.4 41.7 33.4 24.8 16.2 6.1 - 7.1 -22.0 -28.5 -31 79.2 78.4 77.5 75.9 73.2 69.6 65.1 59.0 53.4 47.8 42.4 34.3 25.9 17.2 7.5 - 4.4 -16.6 -24.5 -27.6 178 METHODS IN CLIMATOLOGY able, the geographical latitude (<) is that of J. D. Forbes: 4 t = - 17.8 + 44.9 cos 2 (0 - 630')C. This equation is derived from observations only of the northern hemisphere, but is valid, despite this fact, approximately from 40S to 60N. Such an analytical equation is needed for theoretical investiga- tions. For practical purposes, such as calculating anomalies, Table 46 is preferable to an equation. The values in the table 6 are probably the best available. Particularly, the formerly dubious temperatures of the high south- ern latitudes are replaced by those of W. Meinardus, who derived them from the observations of the most recent expeditions. The series of temperatures of January, July, and the year in the table were revised by Meinardus himself in 1936. The centigrade temperatures of the original table appear in this book converted into Fahrenheit degrees for the first time. Naturally, the series of Table 46 should be interpolated for each degree, if used for calculating the anomalies of a great number of places. In view of the great importance of the method of anomalies, Figure 31 has been inserted, ill order to show how much a repre- sentation like this contributes to the understanding of the effect of ocean currents and continentality upon the temperature at the earth's surface. XIV. 6. c. Pleions, Meions Another application of anomalies and isanomals has been given by H. Arctowski in a series of papers. 6 For a portion of the earth's surface, annual averages of an element are known for a normal period. Anomalies are computed for each year of the period in question and represented by maps of isanomals. Regions encircled by a high positive isanomal are called pleions those characterized by high negative anomalies are named antipleions or melons. The method is applicable to every variable element. If temperature is dealt with, the terms thermo- 4 James D. Forbes, " Inquiries about Terrestrial Temperature," Transactions of the Royal Society of Edinburgh, XXII (1859), pt. I, p. 75 ff. 6 See Hann-Siiring, Lehrbuch der Meteorologie, 5th ed., and V. Conrad, " Die klimato- logischen Elemente . . .," Koppen-Geiger, Handbuch der Klimatologie, IB, 123. 6 E.g., UEnchainement des variations climatiques (Bruxelles: Socidt^ Beige d'As- tronomie, 1909); "Zur Dynamik der Klimaanderungen," Met. Zeit. 1914, pp. 417-426; Communications de I'Institut de Geophysique et de Meteorologie de V Universite de Lw6w. See also the valuable and critical review by Alfred Wegener, in Gerland's Beitrdge zur Geophysik, vol. X, "Besprechungen," pp. 298-299. METHODS OF ANOMALIES 179 180 METHODS IN CLIMATOLOGY pleion and thermomeion are also used. An interesting attempt has been made to study the shifting of the pleions and meions from year to year. These efforts might be successful for the purposes of dynamic climatology. (XVIII. 6.) XIV. 6. d. Snow-Cover and Altitude The duration of snow-cover has been represented by the method of isanomals for the eastern Alps and illustrates the influence of a high mountain range extending east and west, upon snow conditions. 7 XIV. 6. e. Duration of Sunshine vs. Elevation A study was made of the distribution of bright sunshine dura- tion in the eastern Alps, by the method of isanomals. 8 The map (Fig. 32) shows that the positive anomalies are crowded on the southern slopes of the Alps, and the negative on the northern. Thus, the method of isanomals again gives the trend of the effective climatic divide between the northern and southern slopes of the Alps. This result is of great consequence for the appreciation of the method of isanomals, and runs parallel with the method of correla- tion, XIV. 6. f. Anomalies of Precipitation The knowledge of the geographical distribution of rainfall is of the greatest importance for agriculture, hydrodynamics, archi- tecture, etc. Conversely, it would be difficult to find a branch of human activity which is not interested directly or indirectly in the amounts of precipitation which one may expect everywhere on the earth's surface. So it is no wonder that innumerable maps of the distribution of precipitation appear in books, encyclopaedias, journals, papers, and pamphlets. The difficulties in drawing such maps have not been overcome. As long as the region for which the precipitation is to be shown is flat, like a table, the use of isohyets, i.e., lines of equal precipita- tion, is more or less satisfactory. If the country is hilly or even mountainous, then difficulties set in, as in the case of temperature. 7 V. Conrad, M. Winkler, "Beitrage zur Kenntnis der Schneedeckenverhclltnisse in den oesterreichischen Alpenlandern," Gerland's Beitrage zur Geophysik, XXXIV (1931), 473. *V. Conrad, Anomalien und Isanomalen der Sannenscheindauer in den oester- reichischen Alpen (Wien, 1938). METHODS OF ANOMALIES 181 O U o> Si G en Q, C s en <rt CD <u -5 S , 'a L 8. O o V O I 182 METHODS IN CLIMATOLOGY The comparison of like with like is not valid. The difficulties are even much greater than with temperature. With this last, there is great trouble with inversions in winter in the mountain valleys. Regardless of this real obstacle, the normal lapse rate (of tempera- ture) does not vary much over the surface of the earth, if great regions are considered and details excluded. For example: The lapse rate in Peru, 16S, is 0.61 C/100 m; in Ceylon, 7N, 0.64; on the slopes of Mount Etna (Sicily), 38N, 0.64; on Ben Nevis (Scotland), 57N, 0.67 C/100 m. For precipitation, the conditions are of an entirely different nature. For example: for Christmas Island (10S, Indian Ocean, 190 miles south of Java) the author calculated the increase of rain- fall to be 140 mm /1 00 m. On the slopes of the volcano Tengger (Java), C. Braak found 97 mm/100 m, and that a decrease, between 600 m and 1700 m altitude. On the slopes of Mount Idgen (Java), an increase of 260 mm/100 m was stated. In a small valley a few miles long in the European Alps, the increase of precipitation, even with differences in height up to more than 7000 feet, varies from 14 mm/100 m to 163 mm/100 m (0.17 in./lOO feet to 1.94 in./lOO feet). In every mountain range of temperate latitudes a decrease of precipitation with increasing altitude occurs. Another striking example of these conditions is presented by O. Liitschg. 9 In the small region of the famous Mont Cervin (Matterhorn), the following variations of precipitation with height are observed : Variation of Precipitation with Height Pair of Stations mm/ 100 m Visp Zermatt increase 9.3 Grachen Zermatt increase 912.0 Visp Grachen decrease 5.8 Visp Saas Fee increase 20.8 Saas Tamatten Saas Fee invariance In the Rhone valley (Wallis), the conditions mentioned are even more exaggerated : 9 Otto Ltitschg, " Ueber Niederschlag und Abfluss im Hochgebirge," Verd/entlichung. der hydrologischen Abteilung der Schweizerischen Meteor ologischen Zentralanstalt in Zurich (Ztirich, 1926), p. 137). METHODS OF ANOMALIES 183 Variation of Precipitation with Height Pair of Stations mm /1 00 m Martigny-Ville Sion increase 347 Sion Sierre decrease 200 Sierre Visp increase 84 Reckingen Oberwald increase 2252 Oberwald Gletsch increase 46 In view of these conditions no reduction to a fixed level is possible. Therefore, the usual rain map is the representation of actual isohyets. What was said about actual isotherms is true also for actual isohyets, owing especially to the great irregularity of the phe- nomenon in regard to time and locality. The difficulties become very complex, but mostly in hilly and mountainous regions. Here again, the method of the isanomals is the only one that is able to eliminate the influence of height and to represent the true features of rainfall distribution. This method determines, in this case, too, the position of the climatic divide, even if it is shifted toward the lee side. The reason for this phenomenon belongs to the realm of physical climatology. XIV. 7. PRECIPITATION PROFILES In view of the great difficulties with isohyets, profiles of precipi- tation drawn through the mountain ranges are, at least theoret- ically, a good expedient; theoretically, since the profiles can be drawn successfully only if the observations of a sufficient number of suitably located stations are available. The principal requirement is that there be stations at or close to the points of the profile extremes. If this condition is not satisfied, the drawing of the profile is nothing but a guess, and the profile itself represents fancy more than truth. The stations should not be too scattered on either side of the location of the profile. The network of stations up to the crests and summits of the high mountains is very dense in Switzerland, so an example from the western Alps may be useful. This is reproduced in Figure 33, where Liitschg 10 gives a profile through the Bernese Alps from NW to SE. 10 0. Ltttschg, Ueber Niederschlag und Abfiuss im Hochgebirge (Zurich, 1926), p. 138. 184 METHODS IN CLIMATOLOGY This profile may serve in some points as a pattern. The vertical scale is not so exaggerated as is often the case in such repre- sentations. It is 10.5 times greater than the horizontal. The length of the profile is 278 miles, with 27 stations. On the average, Bern- JungFraujoch-Qeisspfadsee- Domodossola - Pavia - * n Ma ^ pHno FIG. 33. Precipitation profile across the Bernese Alps from NW to SE. (After O. Lutschg) there is one station every ten miles. This ratio does not give the right picture, because, deliberately, the stations are not evenly distributed over the profile. The stations are very dense within the region with greatest differences in height. There, 10 stations are located within 37 miles. In the plain of the Po (Italy), two stations are shown within 60 mil^s. It is usual to blacken the mountain profile. In the figure, it is slightly hatched so that certain levels can be indicated by hori- zontal lines. This gives valuable information. The mountain and the precipitation profiles are drawn sepa- rately: the two scales do not overlap, but are arranged one above the other, with homologous points on the same vertical. At the southern side (right), the profile is bifurcated, showing the amount of rain for the plain on the one hand, and for the hills (about 3000 ft) on the other. The clear marks and lines showing the vertical and the hori- zontal scale, should not be overlooked. This representation is well-planned, indeed. Yet one un- familiar with Swiss orography, looking at the graph, might not understand why the amounts increase so much between the 100 km (see figure 33) and the 150 km point. Here, the profile is influenced by the contour features on either side. Nevertheless, profiles based on sufficient and exact data are an excellent and necessary supplement to the maps of actual isohyets, never satisfactory in mountainous countries. Figure 34 exhibits a profile of snowfall across New England from the New Hampshire coast to Williamstown, Mass. C. F. METHODS OF ANOMALIES 185 FIG. 34. Profile of snowfall across New England from the New Hampshire coast to Williamstown, Mass. (After C. F. Brooks) Brooks's graph n is, on the whole, in good agreement with the foregoing suggestions. Representation of the hypothetical por- tion of the curve by a broken line might be generally adopted. An arrow indicates the direction of the snow bearing winds. XIV. 8. VARIABILITY OF PRECIPITATION AND AMOUNTS OF PRECIPITATION The independent variant is the amount of precipitation; the dependent is the variability. This combination shows how adapt- able the method of anomalies and isanomalies is. The problem was discussed in III. 8. "Charles F. Brooks, "The Snowfall of the Eastern United States," M.W.R., January 1915, vol. 43, p. 9. CHAPTER XV WIND XV. 1. WIND ROSES f "^HE WIND ROSE is defined as "a diagram showing for a I definite locality or district, and usually for a* more or less JL extended period, the proportion of winds blowing from each of the leading points of the compass." l In the simplest form, eight directions are used, and relative frequencies (per cent) should be preferred. The frequency of calms is indicated by a circle in the center of the rose. The radius is proportional to the frequency; or, as on United States pilot charts, a number in the circle shows the per cent of calms. 2 The method is too well known to require further explanation. Sir Napier Shaw constructed a composite wind rose or wind "star" in the utmost detail. 3 This is reproduced in Figure 35. It shows the annual course of the ffequency of three classes of wind velocity for 16 directions. The three steps of velocity are : 1) light winds 2) moderate or strong winds 3) gales Light winds (Beaufort 1 to 3), are represented in the rose by a single line. The second group includes the Beaufort degrees 4 to 7, and is indicated by a double open line. At their ends are small blackened sections which represent the number of gales of force 8 and more. Frequencies are given in per cent. "The indenta- tion of the base line allows us to distinguish the information for the summer half, April to September, for the northern hemisphere, from the two winter quarters, January to March on the left for a spectator looking from the centre and October to December on the right. Small letters A and 5, hardly visible in the reproduction without a glass, mark the ends of the salient." Thus, one "star" 1 Meteorological Glossary (3d ed., London, 1939). 1 For a good example of wind roses and their applications see Glenn T. Trewartha, An Introduction to Weather and Climate (2d ed., New York, 1943), p. 112. 8 Sir Napier Shaw, The Drama of Weather (Cambridge, 1933), p. 165. 186 WIND 187 replaces twelve normal wind roses. Drawing one star may be more troublesome than drawing twelve simple roses, but much printing space is saved. Wind roses are often used to indicate the geo- graphical distribution of wind conditions, and are legible even on a rather small-scale map; but this is not true for the "stars." FIG. 35. The wind "star" (composite wind rose). (After Sir Napier Shaw) For climatological purposes, the data used for wind roses are better given in numerical tables, which take into consideration also the frequency of calms. Here, too, percentages are preferable to absolute frequencies. 4 The horizontal entry of these tables of distribution of wind directions has headings for the eight directions and the calm ; the vertical is used for the months. Such statistics are indispensable. XV. La. Wind Roses for Different Elements Wind roses for different elements are based on the principle mentioned, and are illustrated by the example in Table 47. 6 The 4 See models in R. DeC. Ward and C. F. Brooks, "Climates of North America," K6ppen-Geiger, Handbuch der Klimatologie, pt. J, pp. 234 ff. After A. Wijkander, "Observations mt&>rologiques de 1'expddition arctique Su&ioise 1872-73," Kongl. Svenska Vctenskaps-Akad. Handlinger Bandet 12 t no. 7 (Stockholm, 1875). 188 METHODS IN CLIMATOLOGY TABLE 47. WIND ROSES FOR DIFFERENT ELEMENTS OBSERVED AT MOSSEL BAY, SPITSBERGEN (80N, 15E), 1872/73. Deviations for Temperature (A*C) and Cloudiness (AC%), and Per Cent of Probability of Precipitation (P-Pro %) (After A. Wijkander) Wind direction A*C AC% P-Pro % N -1.8 + 9 33 NE -2.4 4 18 E 2.3 17 8 SE +0.5 -13 5 s +4.9 - 6 6 sw +3.8 + 3 19 w -0.4 + 14 40 NW -2.0 + 16 38 Calms -1.1 -10 11 NOTE: To avoid misunderstanding: The " P-Pro " percentage relates to single wind directions; e.g., there are 33 rainy days out of 100 days with N-wind, and only 5 rainy days out of 100 with SE-wind. The representation clearly shows that a strong tendency toward precipitation exists with northerly and westerly winds, and a small one with southerly and easterly winds. numbers in the columns A/C and AC% denote average deviations for the eight directions and for caliji. " P-Pro " means the average probabilities of precipitation. The method combines the immedi- ately observed surface winds with the synchronous values of the different elements. Wind roses are frequently a good climatic characterization, but do not contribute much to air-mass analysis. The correlation between the momentary surface wind and the life history of an air mass is not well defined. XV. 2. MAPS OF STREAMLINES For a cartographical representation, maps of streamlines seem to offer the best climatological information. Streamlines exhibit the state of air motion at a particular moment, and must be carefully distinguished from trajectories or actual paths of an air particle. Streamlines reproduce the synoptic picture at a given moment, trajectories the chronological story of an air particle. Maps of streamlines should be accompanied by maps of the synchronous distribution of air pressure. The technique of drawing streamlines includes the considera- tion that the resultant wind direction at a place should be the tangent to the streamline. Besides, physical considerations con- WIND 189 earning the field of pressure and temperature are of primary interest in drawing streamlines. As was stated in Chapter VIII, the observations and publica- tions include either (a) direction and velocity, or (b) only a fre- quency distribution. In the first case, the resultant wind velocity and steadiness (an important characteristic of wind conditions) can be computed for every station. Climatological streamlines represent, of course, average condi- tions. As far as the wind is concerned, a climatological streamline does not mean a reality like a meteorological streamline : the distinc- tion is the same as between an arithmetical mean and an actual value. The arithmetical mean becomes the more characteristic the smaller the variability. At a tropical place with a very small tem- perature variability the monthly mean of temperature is no longer a fictitious number. In wind conditions " steadiness" replaces " variability." The higher the steadiness, the more the average streamline approximates actuality. Therefore, it is of great in- terest to add the representation of steadiness to that of velocity and direction on a climatological map of streamlines. FIG. 36. Streamlines over the Balkans; average for July 1917. (After E. Kuhlbrodt) As an example, average streamlines are given for the eastern Balkans, according to observations made in July 1917. The special method applied to the construction of this map (Fig. 36), has been elaborated by Alfred Wegener, and fulfills apparently all require- 190 METHODS IN CLIMATOLOGY ments. The trend of the streamlines in Figure 36 is interesting, in that the graph clearly shows how the general west-east streaming of the air over the higher latitudes of Europe is deflected to winds A /'v V f S /' ^' / --X-~-j f x \ s i ... .-JTO ^ - \_J^\ JANUARY STREAMLINES NEW ENGLAND FIG. 37a. Streamlines of New England in January. (After V. Conrad) with a northern component; this is more and more strengthened towards south and east. So the map constitutes a representation of the "Etesians." The feathers, marked directly on the stream- lines, indicate the velocity of the wind in relative measure. In the WIND 191 present map, half feathers are equivalent to the weakest winds, and three full feathers indicate a very strong average velocity. (See the region off the coast of Asia Minor.) JULY STREAMLINES NEW ENGLAND FIG. 37b. Streamlines of New England in July. (After V. Conrad) The steadiness is expressed in the following way: 1) If the streamlines are dotted, the steadiness is between 0% and 33% 2) Broken streamlines mean a steadiness of 33% to 66% 3) Full streamlines indicate a steadiness greater than 66% 192 METHODS IN CLIMATOLOGY From the present representation, one can see that the northerly winds of the Aegean Sea are not only the strongest winds but also those with greatest steadiness. XV. 2. a. Streamlines Based Only upon Frequencies of Direction If records of velocities are lacking, average resultant wind directions may nevertheless be computed, as was proved earlier (see VIII. 2). Then, of course, only streamlines can be con- structed; naturally no velocities, no steadiness can be indicated. Even a plain streamline picture is climatically instructive, as is shown in Figures 37 a and b. At first glance, the reader sees the relatively slight shifting, even if climatically decisive, of the average wind from winter to summer. The continental features of winter on the New England coast clearly appear, and the warm, humid air masses which overflow the country in summer are indicated. CHAPTER XVI AIR MASS CLIMATOLOGY AERAGE STREAMLINES (mentioned in the foregoing chapter) are closely connected with the concept of air masses, so that this branch of climatology, rich in prospect, may be discussed here. The methods used in these investigations are similar to the usual statistical procedures. They start from the frequencies of different types of air masses for certain parts of the year. An air-mass calendar taken from the daily weather maps is the foundation. Relative frequencies should be preferred. One of the results is also of special methodological interest: Air masses can be divided into two main groups: continental (c), and maritime (m). The frequency-ratio c/m is obviously a good characteristic of the climate, and perhaps a better index of con- tinentality than the factors of continentality (discussed later on). Table 48 offers two examples of the seasonal ratios c/m, one for western Europe, 1 the other for the eastern United States. 2 TABLE 48. DEGREE OF CONTINENTALITY DERIVED FROM FREQUENCIES OF CONTINENTAL (c) AND MARITIME (m) AIR MASSES (FOR PENNSYLVANIA AND FOR WESTERN EUROPE) Winter Spring Summer Fall PENNSYLVANIA c/m 1.65 1.08 1.02 1.07 WESTERN EUROPE c/m 0.58 0.40 0.18 0.25 The results of Table 48 do not need further explanation. In the eastern United States, especially in winter, the continental 1 E. Dinies, Luftkorper Klimatologie (Frankfurt a.M., 1932) (air mass system pro- posed by F. Linke). 'H. Landsberg, "Air Mass Climatology for Central Pennsylvania," Gerland's Bcitrage zur Geophysik, vol. 51 (1937) p. 263 (H. C. Willett's classification). For further information see Jerome Namias, Air Mass and Isentropic Analysis (5th ed., with con- tributions by T. Bergeron, B. Haurwitz, G. Miller, A. K. Showalter, R. G. Stone, and H. C. Willett). In this connection, chapter n on "Classification of Air Masses," with list on p. 76, is of particular interest. See, for an informative survey, chapter iv, "Air Masses and Fronts," in Glenn T. Trewartha, An Introduction to Weather and Climate (2d ed., New York, 1943), pp. 190 ff. 193 194 METHODS IN CLIMATOLOGY component predominates; in western Europe, the maritime. The first result is confirmed by the average streamlines in winter over New England (Fig. 37). The contrast between the two climates is well described by the ratio c/m. The great merit of the air mass index of continentality is that it needs no special assumption or hypothesis. Thus, it yields incontestable results. The disadvantage is that the basic air mass calendar entails much work and critical study. CHAPTER XVII NUMERICAL CHARACTERIZATION OF DIFFERENT CLIMATIC FEATURES XVII. 1. CONTINENTALITY f "^HE INCREASE of the annual range of temperature inland is I perhaps the most striking effect of the continental surface J- on climate. Conversely, the annual range (-4) is taken as a measure of the climatic factor, called continentality. Because a general physical correlation exists between annual range and geo- graphical latitude, the range has to be reduced to equality for all latitudes ; thus the expression A sin is the measure sought for. The formula, now frequently used, reads: * C = L7 . X A - 20.4 sin where C is the coefficient of continentality in per cent, A the an- nual range of temperature in centigrade degrees, and < the geo- graphical latitude. Example : 1) Jakutsk (62.0N, 129.7E, 320 ft.) A = 61.6 C 2) Thorshavn (62.0N, 6.8E, 83 ft.) A = 7.6 C Therefore 1.7 X61.6 _ Cjak - ~ 0.88 ' = Cxhor.. = L7 * 8 7 ' 6 - 20.4 = - 6% The second example shows that the constants have to be some- what changed, since a " negative continentality" has no physical meaning. If the continentality is zero, the climate is no longer influenced by continental surfaces in any way. Consequently, it is always better to speak of continentality than of "oceanity." 1 Wl. Gorczynski, "Sur le calcul du degr du continentalisme et son application dans la climatologie," Geografisker Annaler, 1920, p. 324. 195 o 196 METHODS IN CLIMATOLOGY From the methodological point of view, the fact of a negative coefficient is not so important and easily corrected ; the later com- ments by Gorczynski himself and by O. V. Johansson are valuable in this respect. The really weak point, however, has (to the best of the writer's knowledge) never been emphasized, simple as it may be. The coefficient of continentality C becomes infinite (C = <*>), or practically infinite, in the interior tropical belt, if sin <f> 0. Therefore, all these formulae, going back to W. Zenker's investiga- tions, are no longer valid for very low latitudes. This point is fundamental in regard to method, and shows that one must always take into account the boundary conditions of the formulae used. On the other hand, it should not be forgotten that the formula gives a comparably numerical representation of the continentality, although for only a part of the earth's surface. Discussing a formula of continentality given by R. Spitaler and mapped by G. Swoboda, D. Brunt 2 arrives at an equation which shows a clearer physical significance than that mentioned. His formula reads : n e + 0.12 = A//130.61-AS where n e denotes the continentality -factor of a given place, A/ the annual range of temperature, and AS the annual range of the aver- age intensity of solar radiation in the latitude of the place in question. Then, "(n e + 0.12) measures the response in monthly mean temperature to a unit change in the mean intensity of solar radiation." We have only to add that D. Brunt alludes to the point that in the maps of Swoboda no isolines are drawn within the zone 20N to 20S, "on account of the uncertainty in the computation of n e for this region." This remark is in good agreement with the criticism above mentioned. XVII. 2. LIMITS BETWEEN TIMBER-FOREST, STEPPE, AND DESERT. EFFECTIVENESS OF PRECIPITATION Supan and Koppen derive numerical limits of their climatic provinces from average data of temperature and precipitation. For example : 1) The coldest month in the "tropical (A) -Climate" is warmer than 64F. David Brunt, "Climatic Continentality and Oceanity," Geographical Journal, London, vol. LXIV (1924), p. 43. CHARACTERIZATION OF CLIMATIC FEATURES 197 2) The highest average monthly temperature within the 41 regions of snow and frost " (EF-Climates) is below 32F., etc. Supan had already defined the tropical zone by the annual iso- therm of 68 F. This definition marked a great advance in this subject. That was also Koppen's opinion, although his numerical limits are incomparably more practical. Great difficulties arise if the effect of precipitation has to be estimated. Chief of these is the problem of the limit between steppe and timber-forest on the one hand and steppe and desert on the other. The consequences of a given amount of rain are dependent on temperature, owing to the effect of evaporation. It is impossible, either theoretically or practically, to formulate an equation which would take account of all effective elements and factors, such as radiation, wind, soil, continentality, humidity, elevation, etc. Therefore, Koppen had to be satisfied with rough empirical ap- proximations. 3 He himself frequently changed these formulas, so that one example may suffice. Often a slight modification of Koppen's rules helps if a special region has to be described. The effectiveness of precipitation is greater at low tempera- tures. Consequently, for instance, one must discriminate between steppes with precipitation in cool winter and steppes with precipita- tion in hot summer. The limit between a forest climate and a steppe climate is indicated by the following relations between temperature, t (C), and precipitation, r (cm) : a) rainy period definitely in winter: r = 2t b) rainy period definitely in summer: r = 2t + 28 c) no definite rainy period: r = 2t + 14 The limit between steppe and desert can be estimated to lie where the rainfall is one-half of the amount stipulated above, in each case : a) winter rain r = / b) summer rain r = / + 14 c) transition cases r = / + 7 The following example may be instructive : 1 For further details, see W. Ktippen, Grundriss der Klimatologie (2d ed., Leipzig, 1931), and Kflppen-Geiger, Handbuch der Klimatologie, vol. Ic (1936). 198 METHODS IN CLIMATOLOGY 1) Stalingrad (48.7N, 44.5 E, 138 ft):/ = 7.7C;r = 3 2) Achtuba (48.3 N, 46.9E, 11 ft):/ = 7.7C;r = 2S cm 3) Astrachan (46.4N, 48.0E, -30ft):/ = 9.4C;r = 15 cm Astrachan lies in the delta region of the Volga river, Achtuba is 150 miles distant in the Volga basin, and Stalingrad lies another 120 miles up the river. For the present purpose one can assume in rough approximation that the distribution of precipitation over the year is more or less uniform. Therefore the formulas (c) should be valid. For the temperature of Stalingrad and Achtuba we get the formula : steppe climate if r < (2t + 14) = 29 cm Therefore Stalingrad with 37 cm has a forest climate: Achtuba with 25 cm has a steppe climate, and the climate of Astrachan is that of the desert, because there is less than 16 cm rainfall. The boundary between forest and steppe climate runs between Stalingrad and Achtuba. The border line of the desert can be assumed to lie between Astrachan and Achtuba, but closest to Astrachan. An interpolation regarding the location of such a border line between two places should be made with the greatest caution. The reader may be reminded of coherent and non- coherent climatic regions. Every interpolation presumes con- tinuity of the variations. 4 The index of precipitation effectiveness given by C. W. Thorn- thwaite 6 reads : (P \10/9 r^To). The symbol / is called the precipitation-effectiveness index. P means the monthly precipitation in inches and T the average monthly temperature in F, while n indicates the number of the month in question. There is no doubt as to how to calculate this index, but it is a rather wearisome task. The reader who intends to use this formula finds some help from the nomogram given by Thorn- thwaite. 4 Since the stations are generally far distant from one another, the interpolation of the values, critical for the location of the border lines between forest and steppe and between steppe and desert, is not an entirely reliable procedure. Therefore, botanical facts also ought to be considered in drawing these border lines. 5 C, W. Thornthwaite, "The Climates of North America According to a New Classification," Geographical Review, October 1931; " The Climate of the Earth," ibid., July 1933; Atlas of Climatic Types in the United States 1900-1939 (Washington, 1941). PART IV THE CLIMATOGRAPHY CHAPTER XVIII ARRANGEMENT OF A CLIMATOGRAPHY IN THE EARLIER PARTS of this book, methods of examining observational data in a critical and quantitative way were explained. From the homogeneous and reduced series of elements, characteristics were derived by means of simple mathe- matical statistical methods. The idea which runs right through is to proceed from the qualitative to the quantitative, and to arrive at comparable and reproducible results. These are the methods of analysis. But a climatography should give more than this. It should offer a full picture of the climate in question. This aim can rarely be reached. Nevertheless, the content, outlines, and arrangement of a more or less complete climatography are here suggested. XVIII. 1. THE INTRODUCTION The Introduction should contain a short geographical descrip- tion of the region in question, illustrated by a clear, legible, oro- graphic map. At least the trends of rivers, streams, and lakes, as well as those of the mountain ranges, should be outlined. A second map showing the locations of the stations should not be omitted. The reader finds a pattern of the latter in each section of the Koppen-Geiger Handbuch der Klimatologie, e.g., in R. DeC. Ward and C. F. Brooks, "Climates of North America." But a net of parallels and meridians running right across the map should not be forgotten. An alphabetical list of stations is indis- pensable. The three geographical coordinates should be added to each station name, with a number referring to the map of stations. The second part of the introduction should give a survey of the material in connection with the map of stations. The kind of shelters and rain gauges and their exposure at the majority of stations should be briefly and clearly described. The history at 199 200 METHODS IN CLIMATOLOGY least of the " normal stations" should be given, pointing to chang- ings of exposure of the instruments, of methods of observation, of operators, etc. Smoothing of certain series correcting obviously false observa- tions and interpolating of missing values should be indicated. The series whose relative homogeneity has been examined should be given in extenso, as well as detailed results of these com- putations with an exact indication of the methods applied. Simi- larly, a clear report with all necessary numerical additions should be given regarding the reduction to the chosen " normal period." It does not suffice to make a general statement that the series are homogeneous and that they are reduced to a given period. Only from detailed numerical results can the reader judge how far the representation is reliable and useful. XVIII. 2. STATIC CLIMATOLOGY The static climatology deals with the average state of the atmosphere. It is more or less identical with climatological statistics. Therefore, the backbone of this part is represented by material which must be given in numerical tables and graphs. There are different kinds of tables. 1) The climatic table contains average values for a certain place, derived from a certain period, and besides, extreme values. Samples of this kind of table are given in the appendix. The title of each climatic table must show, also, the name of place and country, and the three geographical coordinates. It often happens that it is impossible to give all the data for one and the same period. Then the period must be mentioned at the head of its column, which, in any case, has to contain the units of the respective element. (C, F, tercentesimal degrees or ab- solute degrees, "A," or "K" = Kelvin; inches or millimeters, millibars, millimeters of mercury or inches of mercury, meters, feet, etc.). 2) The climatic table can also be replaced by tables each of which is related to a certain element (average temperature, cloudi- ness, etc.) and contains a list of stations and the averages of the element in question for the months and for the year. As a sample, again the tables in "The Climates of North America" (R. De- Courcy Ward and C. F. Brooks) can be recommended. Many authors prefer tables of this kind because they save space and printing costs. Climatic tables should be given, in any case, for ARRANGEMENT OF A CLIMATOGRAPHY 201 the most representative places of the region. If a climatography of New Hampshire were composed, climatic tables would have to be given for Mount Washington, for a suitable station at its base, and for at least one station near the sea. 3) For stations with more than ten years' operation, tables should be included which contain the monthly and annual means or sums, at least, for temperature, rainfall, and pressure, for all available months and years. Cloudiness should be added, if in any way possible. 1 Whatever kind of climatic tables is considered, frequency dis- tributions (see II. 4 and ff.) of the various elements contribute essentially to the quantitative description of climate and can be advantageous in many even not specifically climatological prob- lems. Recently J. Namias (" Construction of 10000-Foot Pres- sure Charts Over Ocean Areas/' BAMS vol. 25, 1944, p. 177 ff.) has used frequency distributions of air temperature by 5 squares over the Atlantic Ocean for his forecast technique. XVIII. 3. MONOGRAPHS Monographs are of great value, inasmuch as they deal with long homogeneous series of different elements obtained at a place representative of a greater region. It is impossible to give specific instructions for composing such investigations and publications. The more details and statistics are given, the more useful the monograph is. The reader may be referred to some models of different kinds: 1) Under the direction of Dr. J. B. Macelwane, S.J., there appeared Meteorological Means and Normals from Observations at Saint Louis University, ipu-ipJ5, by John J. Renk, S.J., two volumes which offered only numerical tables, extremely useful for other scholars. 2) A. Wagner, in Der Jahrliche Gang der meteor ologischen Ele- ments in Wien (Wien, 1930), presents a full discussion and a great collection of statistical tables. Both monographs are restricted to the annual course of the elements. 3) F. Steinhauser, in Die Meleorologie des Sonnblicks (Wien, 1938), gives the pattern of a monograph which considers the annual 1 For examples see R. DeC. Ward and C. F. Brooks, " Climates of North America/' in Kdppen-Geiger, Handbuch der Klimatologic, p. 266 ff, or U. S. Weather Bureau, 41 Climatic Summary of the United States," e.g., W. M. Wilson, Section 82, South Central New York, p. 3 ff. 202 METHODS IN CLIMATOLOGY course as well as the daily, in an exact way, by numerical data and their discussion. A different kind of monograph is that of O. L. Fassig, The Climate of Baltimore. It will be mentioned later (XVIII. 7.) XVIII. 4. RECORDS OF SELF-REGISTERING INSTRUMENTS If continuous records are at hand, the daily course of the re- spective elements should be shown. Often, two-hourly values are sufficient. In the first place, the daily variation . has to be re- produced numerically. Graphs are always welcome but do not replace numerical tables. From the tables the reader can make himself a correct graph. The converse is not true. From the continuous records periodic and aperiodic variations have to be derived. The periodic variation is the difference between the highest and the lowest value of the average daily course (perhaps for the period of a month) of the element (pressure, temperature, relative humidity, etc.). The aperiodic variation is obtained from the average daily extremes. In the case of temperature, it can be obtained not only from the records of the thermograph but also from the daily readings of the extreme thermometers. XVIII. 5. CHANGE OF TEMPERATURE AND PRECIPITATION WITH HEIGHT In a mountainous country, the variation of temperature with height has to be carefully considered, and its annual course, in any case, must be presented. Here also numerical tables are preferable to graphs alone. The same is valid for the variation of precipitation with height. If the number of stations is not too small, and there are con- siderable differences in height, a standard temperature-height curve and a precipitation-height curve should be constructed. From these curves, the anomalies of different places are calculated. XVIII. 6. GRAPHICAL REPRESENTATION As far as graphical representation is concerned, it is a matter of taste which and how many Cartesian diagrams are shown. If these appear in addition to numerical tables, they are practical and useful. ARRANGEMENT OF A CLIMATOGRAPHY 203 The geographical distribution of at least the most important elements (pressure, temperature, precipitation, cloudiness) should be represented by maps of isolines, provided the country is not of too great ''relief energy" and the mountains do not occupy too large a portion (see details XIV. 2). For a really mountainous country with a dense network of stations up to great altitudes, attention should be focused on the construction of maps of isanomals. Only these maps are able to reveal the real distribution of temperature and precipitation, and particularly the climatic divides, in a mountainous country. The method of isanomals eliminates the influence of height on the element in question and permits a reasonable interpolation for places where no data are available. The anomalies, together with the standard curve for the region, yield the actual values of the element (see XIV. 2). Maps representing the data of the beginning and the end of well-defined seasons contribute much to the knowledge of a climate. Examples are: season with frost, vegetative period, etc. (See VL 2, 3.) The addition of an exact definition of the season represented should never be overlooked. The frost period, for instance, can be defined in different ways: 1) as the period between the dates when the minimum temperature drops below the freezing point the first and the last time in the cold season. 2) as the period between the dates of the first and the last day with a mean temperature = 32F. 3) as the period between the dates at which the average curve of the annual course passes the freezing point. The vegetative period, also, can be defined variously, as follows: 1) as the interval of time which lies between the dates when the average annual course passes 43 F. 2) as the space of time between the last and the first "kill- ing frost"; this period is called "growing season" in the United States. The number of these examples could be greatly increased. There- fore a clear, exact definition is essential. As we have already pointed out, the dates of the beginning and the end of a period yield the duration of the period. The cartographical representation of "durations" is valuable. Gener- 204 METHODS IN CLIMATOLOGY ally they are calculated in days. For use in maps they should commonly be converted into weeks. These correspond better to the usual degree of accuracy of the data and particularly to that FIG. 38. Isolines of the dates at which the curve of the annual course of temperature rises to 70F in the Mediterranean region. (After V. Conrad) FIG. 39. Isolines of the dates at which the curve of the annual course of temperature drops below 70F in the Mediterranean region. (After V. Conrad) of the trend of the respective isolines. In some cases the unit of a month yields the best and the most reliable isolines. Figures 38 to 41 illustrate the method of representing the dates of beginning and end of defined periods, and the duration ARRANGEMENT OF A CLIMATOGRAPHY 205 of such periods. (See also XIV. 5. a.) 2 In Figure 38, isolines of the dates are seen, on which the curve of the annual course of temperature reaches 70F in the Mediterranean region. These FIG. 40. Isolines of the duration in weeks of an average daily temperature i 70F in the Mediterranean region. (Duration of the hot season). (After V. Conrad) FIG. 41. Isolines of the number of months with an amount of rain ^ 0.2 inch. (Isolines of the length of the dry season, Mediterranean region). (After V. Conrad) isolines are characteristic of the beginning of the hot period; the next, Figure 39, shows its end in the same way. On the two maps, the average migration of the chosen threshold of temperature can 2 S. S. Visher, "The Seasons' Arrivals and Lengths," Annals of the Association of American Geographers, vol. 33, 1943, pp. 129-134, gives similar maps for the United States. 206 METHODS IN CLIMATOLOGY be observed. The average temperature reaches 70F in North Africa on May 5, migrates northward and arrives in southern Europe, June 10. The retarding effect of the water basin itself is well emphasized by means of the closed isoline which corresponds to June 5. 3 Figures 40 and 41 give examples of durations. The first of them is derived from Figures 38 and 39, and represents the dura- tion in weeks of temperatures equal to or higher than 70F. It is the duration of the hot season in this region. On the average, the duration increases from north to south. The intensity of this increase is, however, greatly diminished by the various effects of the surface of the water. The discussion of a map of well drawn isolines can be very stimulating and can result in a surprising insight into the regional climate. The last example for this group of problems is offered by Figure 41, representing the length of the dry season in the Mediterranean region. Three features of this map may be stressed: 1) The "dry season" is well defined by the indication: Number of months with an amount of rain ^ 0.2 in. 2) Parts of the isolines, fhe trends of which are only as- sumed, not really based on observations, are drawn as broken lines. 3) The month is taken as a unit, because of the high variability especially of small amounts of rain, and the great uncertainty in measuring such tiny quantities, particularly at times of high temperature and great evaporation. XVIII. 7. DYNAMICAL CLIMATOLOGY This expression was coined by Tor Bergeron, when he extended the meteorological methods of the Norwegian school to climatol- ogy. 4 Static or statistical climatology represents the average and extreme states of atmospheric conditions. Dynamical climatology means perhaps development of the average states. No doubt many features of each division overlap. Usually, every old- fashioned climatography contains, for instance, variabilities which belong to the dynamic division. 1 This date which belongs to this isoline was unfortunately omitted in the figure. 4 See review by H. C. Willett, "Ground Plan of a Dynamic Meteorology, MWR Vol. 59, (1931), 219-223. ARRANGEMENT OF A CLIMATOGRAPHY 207 208 METHODS IN CLIMATOLOGY ARRANGEMENT OF A CLIMATOGRAPHY 209 210 METHODS IN CLIMATOLOGY ARRANGEMENT OF A CLIMATOGRAPHY 211 A forerunner of these later efforts and their realization is repre- sented by the excellent study of Oliver L. Fassig, The Climate of Baltimore.* The first part of Fassig's work deals with " static climatology," the second with "dynamical climatology." There the " types of weather characteristic of the geographical horizon of Baltimore" play an important role. A large series of typical weather maps on the occasion of special storms, rain areas, frost areas, cold waves, blizzards, are published in this highly interesting book, which may serve as a pattern for future climatographies. Figures 42 to 45 represent an interesting attempt. The average distribution of pressure, wind, and temperature of certain months (December, March, June, October) which are classified as cold, normal, or warm, is shown in the maps. The four months are chosen as representative of the four seasons. Although a discus- sion of the maps would be out of place here, it is clear from these charts that this is the way to understand the development of the climatic features of a region. From these attempts, it is only one step to the numerical and graphical representation of average weather types: e.g., 20 cases of Strong foehn wind, east of the Rocky Mountains, are chosen. " Strong foehn" has to be defined, for example, by the condition that the temperature rises 10F or more within one hour at a certain place, representative of the phenomenon. High floods, blizzards, hot and cold spells, dry and wet spells can be satisfactorily dis- cussed by means of average numerical values and their carto- graphical representation. In the writer's opinion, such dynamical methods would arrive at the best physical synthesis of the climatic elements. In this way, a description of important and character- istic climatic phenomena could be given for great regions of the continents and for some parts of the oceans, as a guide for long range forecasting 6 and for many other purposes. 6 Oliver L. Fassig, The Climate of Baltimore (Maryland Weather Service, Special Pub., vol. 2, Baltimore, 1904). See also Gardener Reed, Jr., in QJRMS 1910, p. 39. 6 The ideas of a dynamic climatology have also been well expressed in some studies by C. F. Brooks and his pupils in " Papers in the Relation of the Atmosphere to Human Comfort' 1 (MWR, vol. 53, 1925, pp. 423-437), containing the following articles: C. F. Brooks, "The Cooling of Man under Various Weather Conditions"; E, C. Donnelly, "Human Comfort as a Basis for Classifying Weather"; F. Howe, "The Summer and Winter Weather of Selected Cities in North America"; E. S. Nichols, "A Classification of Weather Types"; I. E. Switzer, "Weather Types in the Climates of Mexico, the Canal Zone, and Cuba." See also E. S, Nichols, "Frequency of Weather Types at San Jose*, Calif.," MWR, vol. 55 (1927), pp. 403, ff. The investigations mentioned attempt to avoid the very complicated system of E. E. FedoroflF (MWR vol. 55, 1927, pp. 401 ff.) and consider the fact that the principles of the representation should not be more com- plex than the phenomenon itself. 212 METHODS IN CLIMATOLOGY XVIII. 8. THE BIOCLIMATOLOGICAL PART A bioclimatological section although a short one is expected in a climatography. Radiation, cooling power, drying power should be mentioned. If no direct observations exist, formulae (mentioned above) enable the climatographer to give estimations of cooling-and drying-power by means of average values of temperature, wind, and humidity. If direct observations of radiation are lacking, at least records of duration of bright sunshine and descriptions of radiation-climate or personal experiences permit a first estimation Of radiation con- ditions. Some words about acclimatization, endemic diseases, health resorts, etc., are usually desired in a more or less complete climatog- raphy. The immense hygienic literature presents reliable sources from which the climatographer can obtain good information with- out assuming too much responsibility. Every climatologist should be cautioned against giving his own opinions about subjects which are outside the range of his special realm. 7 On the other hand, personal experiences with reference to human comfort and striking phenomena in the organic and inorganic realm are always of the greatest value and should not be suppressed. XVIII. 9. DESCRIPTION OF CLIMATIC PHENOMENA It should never be forgotten that geography, and even more climatology, started with descriptions by early travelers. From the books of travels of the famous Venetian Marco Polo to those of Alexander von Humboldt, a nearly innumerable series of keen travelers with open eyes gave us our first knowledge of the climate of remote countries. Nor should modern climatography renounce the lively description of climatic impressions drawn from personal experience. Puka, Albania (42.0N, 19.9E, 2800 ft) is located on the steep slopes of the Albanian Alps, which run along the Adriatic Sea coast from about NW to SE. Here, in summer, an average of six inches of rain falls in eight rainy days. The rain intensity is, on the average, about 0.8 in. (For comparison, in Boston the intensity is 7 An instructive example of this restriction to an objective report (as far as non- geophysical influences and effects are concerned) is offered by B. Gutenberg, "Geo- physik and Lebewesen," in Lehrbuch d. Geophysik, edited by B. Gutenberg (Berlin, 1929). FIG. 46. Eroded mountain slopes with westerly exposure above Puka, Albania. (42 X, 19.9 K, 2800 ft) ARRANGEMENT OF A CLIMATOGRAPHY 213 0.3 in.) This numerical description is interesting and character- istic of the mountain climate along the Adriatic coast. A glance at Figure 46 gives life to the numbers, and we see the effect of these downpours separated from one another by long droughts. The slopes with westerly exposure are deeply eroded; desertlike, barren ground is before our eyes, and that in a climate where olive, fig-tree, and pomegranate thrive and flourish in well protected sites. Thus the climatographer should not omit descriptions and suit- able pictures which offer the means of making his statistical and dynamical features impressive. Books of travels, guidebooks, etc., often provide good contributions in this respect, if personal ex- perience is lacking. Tree growth and vegetation are in closest relation to climate, and architecture also. In the Saas Fee valley in the south of Switzerland (one of the most beautiful valleys of the Alps), the hay barns stand on high stone pillars. Even if observations of precipitation did not exist, from the height of the pillars we should know not only that copious snowfalls occur in this region but also the average maximum depth of snow on the ground. In the whole Mediterranean region from Italy to North Africa, narrow lanes are usual so narrow that two people can hardly pass one another. High altitudes of the sun, combined with frequently cloudless skies and a dry, clean atmosphere with low turbidity factors characterize the weather of the summer half-year. The architectural effect resulting from these scientifically described radiation conditions is narrow lanes. Conversely, we conclude that high radiation is present when narrow lanes are the charac- teristic architecture. Arcades along the streets signify frequent downpours as well as intense radiation. High gable roofs are often the architectural expression of habitually intense snowfalls. High walls encircling precious orchards and vineyards are characteristic of the northern coast of the Mediterranean, protect- ing the crops against the mistral and the bora and similar bad and frequent storms. In Normandy and Brittany the traveler is surprised to see miles and miles of high hedges, which surround fields and farms. One concludes, here again, that frequent storm- like winds are found where such walls and hedges are typical. 8 Numerous settlements high on the slopes and a deserted bottom of the valley indicate frequent floods, and also intense inversions 8 The reader will find excellent examples of climatically characteristic photos in the Atlas of American Agriculture, in the sections on "Soil" and "Natural Vegetation." 214 METHODS IN CLIMATOLOGY of temperature in fall and spring. Manners and customs of man- kind are often highly influenced by climatic features. It was natural that the traveled R. DeCourcy Ward was enthusiastic about the necessity of colorful descriptions; he went so far as to speak of a " car- window climatology." None other than J. Hann always enjoyed getting lively descriptions to cover his numerical skeletons. The many descriptions in his famous regional climatog- raphy of the world attest these efforts. CONCLUSION THE READER who runs over the pages of this book is perhaps disappointed in not finding this or that special method. In the first pages, it was shown that the number of de- rived elements is not limited theoretically; thus completeness is unattainable. 1 Moreover, it is intended to present only a system of methods which facilitates the step from the qualitative to the quantitative. Many ways are offered for attaining this purpose, and they all should lead not only to an exact and comparable description of the climate but, in the end, to a physical explanation. 1 The reader will find a rich source of different methods of representation in the monumental work: Atlas of American Agriculture: Physical Basis including Land Relief, Climate, Soils, and Natural Vegetation of the United States (Washington, 1936), prepared under the supervision of O. E. Baker, with contributions from the Weather Bureau, Willis R. Gregg, H. G. Knight, Frederick D. Richey, F. A. Silcox, A. G. Black. 215 APPENDIX I FORMS OF INDIVIDUAL CLIMATIC TABLES THE FOLLOWING form is that of the " Climatic Summary of the United States" published by the U. S. Weather Bureau. The form is slightly modified with respect to its heading. Two columns have been added: " Depth of Snow on the Ground " and " Estimated Average Wind Velocity." The first columns can be filled with informa- tion, based on the regular observations of cooperative stations in the United States. The average wind velocity is not recorded by the cooperative ob- servers. An estimate of the average of the whole day could easily be made according to the following simplified scale: = calm to light 1 = gentle 2 = strong to stormy Too little attention is paid to the wind in climatological publications. BLANK FORM OF A CLIMATIC TABLE (Taken from the "Climatic Summary of the United States," U. S. Weather Bureau) Name of the Place Country < == X = H = feet (meters) Period: [Remarks about special local conditions, period, instruments, etc.] Temperature Precipitation Relative Humidity Sun- shine Wind 8 ^ Snow E s jJJ -M .S CO S to 3 o W -H *<H flj c **^ rt rt S bo ca O Q ^ J5 js O t? c s 1 <D 3 *O 0) f 1 Absolute Maximum 1 Average of the Minim Absolute Minimum Si jE: CO 4> ,bf) Lowest Monthly Aver 1 1 1 Number of Days with 0) rS 1 1 (U s c A Average Amount Greatest Amount in 2 1 O o S ex Q E r ^ f c i 0> i 1 Average Hours Percentage of Possible Prevailing Wind Direc Estimated Average W * According to daily estimations: calm to light, 1 ** gentle, 2 = strong to stormy. 216 APPENDIX I 217 Julius Harm suggested the following content of a climatological table. The form is in principle identical with that of the U. S. WeatherBureau. A great number of climatologists, perhaps the majority, have made use of the following suggestions as far as available data permitted. A. Temperature a. Averages for the three observation times b. True monthly means c. Extreme monthly and annual means for the period in question d. Averages of the interdiurnal variability e. Mean daily periodical range (if records of a thermograph are available) f. Mean daily extremes g. Mean absolute extremes and their difference h. Absolute extremes i. Average frequencies of certain differences between consecu- tive daily means (e.g. differences: 3F to 6; 7 to 10F etc.) B. Vapor pressure C. Relative humidity (morning, early afternoon, daily mean) D. Cloudiness (morning, early afternoon, daily mean) E. Sunshine duration (hours, percentage) F. Precipitation a. Average amounts b. Relative variability c. Snow (and snow mixed with rain), amounts d. Depth of snow on ground G. Wind a. Resultant direction b. Resultant velocity (vectorial) c. Average velocity, disregarding the direction d. Steadiness H. Air pressure a. Averages b. Average monthly and annual range I. Number of days a. Clear. Overcast b. Without sunshine c. With precipitation S d. With hail e. With snowfall f. With thunderstorm g. With storm ( > Beaufort 6) h. With frost (minimum below freezing point) i. With fog This table should be completed at least by a frequency distribution (per cent) of 8 wind directions and calm for the months and the year. Other frequency distributions could be added according to the purpose of the climatography. APPENDIX II (See XII.2.b and the "Intermediate Chapter") 1 234567 1.000 .707 .577 .500 .447 .408 .378 .354 .333 10 .316 .302 .289 .277 .267 .258 .250 .243 .236 .229 20 .224 .218 .213 .209 .204 .200 .196 .192 .189 .186 30 .183 .180 .177 .174 .171 .169 .167 .164 .162 .160 40 .158 .156 .154 .152 .151 .149 .147 .146 .144 .143 50 .141 .140 .139 .137 .136 .135 .134 .132 .131 .130 60 .129 .128 .127 .126 .125 .124 .123 .122 .121 .120 70 .120 .119 .118 .117 .116 .115 .115 .114 .113 .113 80 .112 .111 .110 .110 .109 .108 .108 .107 .107 .106 90 .105 .105 .104 .104 .103 .103 .102 .102 .101 .101 100 .100 .100 .099 .099 .098 .098 .097 .097 .096 .096 110 .095 .095 .094 .094 .094 .093 .093 .092 .092 .092 218 APPENDIX III AUXILIARY TABLE, CALCULATING THE EQUIVALENT TEMPERATURE FROM AIR-TEMPERATURE (/C) AND RELATIVE HUMIDITY (AFTER F. LINKE) (See IX.2) K(t t b) Air-Pressure in Millimeters of Mercury *C 780 760 740 720 700 -45 0.11 0.11 0.12 0.12 0.12 -40 0.20 0.20 0.21 0.22 0.23 -35 0.33 0.34 0.34 0.35 0.36 -30 0.57 0.59 0.61 0.62 0.64 -25 0.95 0.97 1.00 1.02 1.05 -20 1.55 1.59 1.64 1.68 1.72 -18 1.87 1.92 1.98 2.04 2.09 -16 2.29 2.34 2.40 2.47 2.52 -14 2.75 2.81 2.88 2.97 3.02 -12 3.26 3.37 3.45 3.54 3.61 -10 3.90 4.01 4.11 4.23 4.33 - 8 4.72 4.87 5.00 5.12 5.26 - 6 5.80 5.87 6.00 6.20 6.40 - 4 6.77 6.90 7.13 7.33 7.60 - 2 7.90 8.00 8.25 8.50 8.75 9.09 9.32 9.58 9.84 10.1 2 10.5 10.7 10.9 11.2 11.5 4 12.0 12.2 12.5 12.9 13.3 6 13.9 14.2 14.6 15.1 15.5 8 15.8 16.2 16.6 17.2 17.6 10 18.1 18.6 19.1 19.6 20.1 12 20.6 21.1 21.7 22.3 22.9 14 23.4 24.2 24.8 25.5 26.2 16 26.6 27.3 28.2 28.9 29.7 18 30.2 30.8 31.7 32.7 33.6 20 34.2 35.1 36.1 37.0 38.2 21 36.4 37.2 38.4 39.4 40.4 22 38.6 39.5 40.6 41.7 43.0 23 40.8 41.9 43.4 44.5 45.6 24 43.5 44.5 46.0 47.3 49.4 25 46.7 47.5 48.8 50.2 51.6 26 49.0 50.1 51.6 53.0 54.6 27 51.9 53.2 54.6 56.9 57.8 28 54.9 56.3 57.9 59.6 61.2 29 58.3 59.7 61.4 63.0 64.7 30 61.9 63.5 65.2 67.0 68.8 31 65.4 67.2 69.1 71.1 72.6 32 69.2 71.2 72.9 75.2 76.7 33 73.3 75.2 77.1 79.3 81.3 34 77.6 79.6 80.4 84.0 85.0 35 82 84 86 89 91 36 87 89 91 94 96 37 92 94 96 99 102 38 97 100 102 105 108 39 103 105 108 111 114 40 108 110 113 116 120 219 APPENDIX IV THE DAYS OF AN ORDINARY YEAR, NUMBERED CONSECUTIVELY, BEGINNING JANUARY FIRST Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. 1 1 32 60 91 121 152 182 213 244 274 305 335 2 2 33 61 92 122 153 183 214 245 275 306 336 3 3 34 62 93 123 154 184 215 246 '276 307 337 4 4 35 63 94 124 155 185 216 247 277 308 338 5 5 36 64 95 125 156 186 217 248 278 309 339 6 6 37 65 96 126 157 187 218 249 279 310 340 7 7 38 66 97 127 158 188 219 250 280 311 341 8 8 39 67 98 128 159 189 220 251 281 312 342 9 9 40 68 99 129 160 190 221 252 282 313 343 10 10 41 69 100 130 161 191 222 253 283 314 344 11 11 42 70 101 131 162 192 223 254 284 315 345 12 12 43 71 102 132 163 193 224 255 285 316 346 13 13 44 72 103 133 164 194 225 256 286 317 347 14 14 45 73 104 134 165 195 226 257 287 318 348 15 15 46 74 105 135 166 196 227 258 288 319 349 16 16 47 75 106 136 167* , 197 228 259 289 320 350 17 17 48 76 107 137 168 198 229 260 290 321 351 18 18 49 77 108 138 169 199 230 261 291 322 352 19 19 50 78 109 139 170 200 231 262 292 323 353 20 20 51 79 110 140 171 201 232 263 293 324 354 21 21 52 80 111 141 172 202 233 264 294 325 355 22 22 53 81 112 142 173 203 234 265 295 326 356 23 23 54 82 113 143 174 204 235 266 296 327 357 24 24 55 83 114 144 175 205 236 267 297 328 358 25 25 56 84 115 145 176 206 237 268 298 329 359 26 26 57 85 116 146 177 207 238 269 299 330 360 27 27 58 86 117 147 178 208 239 270 300 331 361 28 28 59 87 118 148 179 209 240 271 301 332 362 29 29 88 119 149 180 210 241 272 302 333 363 30 30 89 120 150 181 211 242 273 303 334 364 31 31 90 151 212 243 304 365 220 INDEX Abbe, E., 134, 135, 136, 137, 138, 140 "Above normal," 37 Absolute degrees of temperature, 200 Absolute drought, 123 Absolute extremes, 17, 30 Absolute frequency, 19 Absolute maximum, 19, 28 Absolute minimum, 19, 28 Absolute probability of precipitation, 53 Absolute temperature, 200 Absolute variability, 31 Absolutely homogeneous climatological series, 130 ff. Acclimatization, 212 Actual isohyets, 183 Actual temperatures, 172 Air mass, continental, 193 index of continentality, 193 maritime, 193 Air mass analysis, vs. windroses, 188 Air mass climatology, ix, 193 Air temperature, "true," 13 Amplitude, relative, 78 Angot, A., 96, 121 Angstrom, A., 115 Annual course of the amounts of precipi- tation, 120 Anomalies, methods of, ix, 170, 173 ff. of precipitation, 180 Antipleions, 178 Aperiodic variations, 202 Aperiodical range of temperature, 88, 89 Arctowski, H., 178 Aridity, index of, 127 Arithmetical mean, 30 Arkin, H., 22, 45, 46, 87 Array, 24 Artificial heating, 94, 95, 97 Ash, D. H., 113 Aspiration-thermometer, 13, 14 Association, degree of, 151 Asymmetry of the annual course of tem- perature, 99 Asymmetry of frequency curves, 44 ff. Atmospheric pressure, 100 Average absolute extremes, 30 Average variability, 31, 46 Azimuth of wind direction, 103 Bacon, F., viii Baker, O. E., 215 Bavendick, F. J., 32 Beaufort, Sir Francis, 186 Beaufort numbers, equivalent of, 105 Bell-shaped curve, 34 Bergeron, T., 193, 206 Berlage, H. P., Jr., 102 Best, A. C., 14 Biel, E., 51 Bigelow, F., 84, 89, 90, 121 Bilaterally limited variates, 44 Binomial coefficient, 69 Bioclimatological part of a climatography, 212 Birkeland, B. J., 147 Bjerknes, V., 169 Black, A. G., 215 Blair, T. A., 97 Blue Hill Observatory, x, 13, 88, 116, 117 Book of normals, 51 Bottom layers, 15 Braak, C., 182 Break of series, 12 Brightness of the sky, 57 British Meteorological Office, 7, 8 Brooks, C. E. P., 87, 88, 89, 160, 161 Brooks, C. F., 13, 15, 46, 84, 95, 113, 114, 122, 185, 199, 200, 211 Bruckner, E., 147 Bruckner-cycle, 147 Brunt, D., 21, 78, 114 Brunt's continentality factor, 196 Brunt's criterion, 78 Buttner, K., 113 Calms, 105 "Car window climatology," 214 Central heights of the class intervals, 19 Central Institute (Office), 7 Change of signs, 134 Chapman, F. H., 37 Characteristics, elementary, 17, 24 system of, 17 higher, 17 primitive, 17 221 222 METHODS IN CLIMATOLOGY Class intervals, 19 Clayton, H. H., 8, 132 Clear days, 115 Climate, definition, 1, 2 Climatic divide, ix, 149, 183, 203 Climatic elements, synthesis of, 211 Climatic exposure, 176 Climatic limits of timber forest, steppe, desert, 196 Climatic phenomena, description of, 212 Climatic regions, coherent, ix, 148 incoherent, 148 Climatic tables, 216 Climatography, 1, 199 arrangement of, 199 bioclimatological part, 212 Climatological averages, reduction to a certain period of, 139 Climatological elements, combined, 4 derived, 4 groups of, 3 primitive, 4 Climatological normals, 148 Climatological series, absolutely homo- geneous, 131 ff. comparableness of, 3 definition of the relative homogeneity of, 134 discontinuities of, 12 Climatology, 1 air-mass, ix, 193 dynamical, 206 ff. static, 200 Climogram, 107 Cloudiness, 115 a bilaterally limited element, 44 and relative sunshine duration, 55, 115 reduction of, to the "true mean," 91 Cloudy days, definition, 115 Codex, International Meteorological 83 Coefficient, binomial, 69 of correlation, 151 of variation, 38 Coherent climatic regions, ix, 148 Cold days, spells of, 95 Colton, R. R., 22, 45, 46, 87 Combined elements, 4, 107 Committee, International Meteorological, 6 Comparableness of Climatological series, 2,3 Comparison, graphic, 166 " Computer's Handbook/' 160 Conferences, International Meteorological, 6 Conrad, V., 2, 5, 13, 37, 38, 49, 50, 51, 52, 57, 59, 61, 62, 64, 65, 75, 91, 92, 97, 99, 100, 104, 113, 124, 126, 134, 135, 136, 138, 144, 146, 148, 149, 174, 176, 178, 180, 181, 190, 191, 204, 205 Constituents of a wave, 77 Continental air masses, 193 Continentality, ix, 4, 195 index of, based on air mass frequency, 193 Continentality factor, Brunt's, 196 Convergence of series, 78 Cooling power, 112 ff., 212 new formulas for, 1 13 and physiological processes, 114 Coolings, 49 Cooperative observers and stations, 9,11, 15, 17, 87, 115, 125 Cornu's theorem, 39 Correction to the "true mean," 88, 89, 90 Correlation, ix, 150 ff. example of calculating, 153 linear, ix, 150 partial, 161 simplifications in computing, 155 Correlation coefficient, 151, 164 partial, 164 Crestani, G., 92 Cumulated temperatures, 96 Curve fitting, 58 Curve parallels, 160 Czuber, E., 19 Daily course of temperature, 81 Daily mean, 83 Decay curves, 62 Deciles, 24 Degree of association, 151 Degree days, 97 Degrees absolute, 200 Deluc, J. A., 1 Denison, F. R., 15 Departure, 31 Derived elements, 4 Description of climatic phenomena, 212 ff. Desert and steppe, climatic borderline between, 196 ff. Deviation, 31 Differences, method of, 139 quasi-constancy of, 131 Dines, W. M. ( 160 Dinies, E., 193 Direction of wind, 103 Directional quantities, 48 Discontinuities of Climatological series, 12 Dispersion of a variate, 38 INDEX 223 Distribution, normal, 31 of temperature in mountainous regions, 170 Divides, climatic, 149, 182, 203 Donnelly, E. C, 113, 114, 211 44 Dot charts," 153, 160 44 Dot diagram,' 1 157 Drought, absolute, 123 partial, 124 Dry season, 205 Dry spells, 123, 124 Drying power, 111 Duration, 94 cart pluviometrique relatif, 120, 121 Effectiveness of precipitation, 196, 198 index of, 198 Ekhart, E., 126 Ekholm, N., 87 Elementary characteristics, 17, 24 Elements, combined, 4 derived, 4 primitive, 4 Endemic diseases, 212 Ephemerides Societatis Meteorologicae Palatinae, 5 Equivalent temperature, 4, 108 ff. auxiliary table calculating the, 219 Equivalents of Beaufort numbers, 105 Eredia, F., 107 "Etesians," 190 Evaluation of Fourier's series, 76 Excess, 38, 41 positive, 38 Exner, F. M., 162 Exponential curve, 63 Exposure, climatic, 176 of the thermometer shelter, 15 41 Extremely above normal," 37 41 Extremely subnormal," 37 Extremes, absolute, 17 average absolute, 30 mean daily, 30 of temperature, 87 Fassig, O. L., 202, 207, 208, 209, 210, 211 Favorable events, 35 FedorofT, E. E., 211 Fergusson, S. P., 118 Fisher, L. C, 32 Fisher, R. A., 21, 29,41,46 Fog, 116 Forbes,]. D., 178 11 Force majeure," 36 Forecast, 149 Fourier's series, 70, 78, 79 evaluation of, 76 Frequencies of wind directions, 103 Frequency, absolute, 19 relative, 19 Frequency curve, 21, 23 asymmetry of the, 44 ff. Frequency distribution, 19, 201 Frogner, E., 147 Frost days, 95 Frost period, 4, 93, 94, 95 different definitions, 203 Galilei, Galileo, 1 Gal ton, F., 160 Gauss, K. F., 34 Gaussian distribution, 33 Gaussian law, 147 Geiger, R., 14, 199 General equation of a conic section, 64 Geographical latitude and temperature, 177 Germination, temperature of, 94 Gherzi, E., 53, 127 Gotz, F. W. P., 4, 107 Goodnough, X. H., 132 Gorczynski, W., 127, 195, 196 Graphic comparison, 166 Graphic methods of representation, ix, 202 "Greatly above normal," 37 41 Greatly subnormal," 37 Gregg, W. R., 215 Grouping the data, 19 Growing season, 95, 203 Gutenberg, B., 212 Hand, I. F., 167 Hann, J. von, viii, 80, 124, 145, 149, 171, 177, 178, 179, 214, 217 Harmonic analysis, 70 ff. Pollak's tables to, 73 Haurwitz, B., 193 Health resorts, 212 Heat balance of the human body, 114 Hellmann, G., 7, 52 Helmert, F. R., criterion, 134, 135, 138, 140 Higher characteristics, 17 Hildebrandsson, H. H., 7 Hill, L., 112, 113 Histogram, definition, 21 of rain, 127 Homogeneity, 200 relative, of climatological series, defi- nition, 134 Homogeneous climatological series, 130, 131 ff. Hot days, spells of, 95 224 METHODS IN CLIMATOLOGY Hot season, length of, 205 Howe, F., 211 Human body, heat balance of the, 1 14 Human skin, temperature of, 1 14 Humboldt, A. von, 212 Huntington, E., 153 Hyperbola, different forms, 64 Ice days, 95 spells of, 96 I MC = International Meteorological Com- mittee, 84 Incoherent climatic regions, 148 Index, of aridity, 127 of continentality, based on air masses frequency, 193 of precipitation-effectiveness, 198 Inhomogeneity of climatological series, 130 Integration, numerical, 82 Intensity of rain, 54, 122 Interannual variability, 50 Interdiurnal variability, 48 Interhourly variability, 50 International Climatological Register, 10, 81 International Conference of Maritime Meteorology, 6 International Meteorological Codex, 7, 83 t International Meteorological Committee, 6, 83, 84, 105 International Meteorological Conference (Utrecht, 1923), 8 International Meteorological Congress (Vienna, 1873), 6 International Meteorological Organiza- tion, 14 (Utrecht, 1874), 9 Interpolation of missing observations, 148 Interseasonal variability, 50 Intersequential variability, 50 Introduction to a climatography, 199 Inversion of temperature, 13, 65, 170 Isanomals, ix, 126, 173 ff. maps of, 203 method of, 174 Isogram, 166 ff. Isohels, 167 Isohyets, 167, 180 actual, 183 Isolines, maps of, 166 ff. projections for maps of, 169 rules for drawing, 167 Isonephs, 167 Isopleths, 166 Iso-stereograms, 167 Isotherms, 167 Johansson, O. V., 196 Kelvin, Lord, 200 Killing frost, 31, 95, 203 Kincer, J. B., 9 Knight, H. G., 215 Knoche, W., Ill Koppen, W., 46, 53, 54, 99, 115, 196, 197, 199 Kubitschek, O., 64, 126 Kuhlbrodt, E., 189 Kurtosis, 41 Lambert's formula, 101, 102, 104 Landsberg, H., 193 Lapse rate, uniform average of, 170, 171, 172, 182 Latitude, geographical, and temperature, 177 Least squares, method of, 60, 65, 71, 87 Left skewness, 44 Lehmann, H., 113 Linear correlation, ix, 150 ff. Linke, F., 108, 109, 219 London, Meteorological Office, 51 Long-range forecast, 2, 149 Lower quartiles, 28 Lutschg, O., 182, 183, 184 * McAdie, A., 84, 117 Macelwane, J. B., 201 Mahalanobis, P. C., 38 Manila, Observatorio Central de, 7 "Mannheimer Akademie," 5 Map of isolines, 166 ff. Mapping temperature conditions of a mountainous region, 170, 171 Maritime air masses, 193 Martonne, E. de, 127 Maurer, J., 171, 174 Maximum, absolute, 19, 28 Mean, weighted, 85 Mean daily extremes, 30 Mean departure, 31 Mean deviation, 31 Median, 14 Meinardus, W., 166, 171, 177, 178 Meions, 178 MSL = mean sea level, 8 Meteorological Codex, International, 83 Meteorological Committee at Vienna (1927), 105 Meteorological Glossary (London), 1, 31, 105, 123, 148, 152, 186 Meteorological Office (London), 51, 148, 152, 156 INDEX 225 Meteorological Register, 9, 17 Methods, of anomalies, 170, 174 of climatological observations, 12 of differences, 139 of least squares, 60 of quotients, 142 semi-average, 61 Meyer, H., 79, 115 Microclimatology, 14 Midpoint of the class interval, 19 Miller, G., 193 Miller, J. K., 169 Minimum, absolute, 19, 28 Missing observations, interpolation of, 148 Modal class, definition of, 21, 22 Mode, 21 Morikofer, W., 4 Mohn, H., 166 Monographs, 201 Monotony of high temperatures, 21 Months of equal length, reduction to, 120 Mountainous regions, mapping tempera- ture conditions of, 170, 171 change of precipitation with height in, 202 Mount Washington Observatory, 18, 22, 30,49 Namias, J., 193, 201 Network of stations, 5, 6 Neuberger, H., 62, 118 Nichols, E. S., 211 Nipher, F. E., 15 Non-coherent climatic regions, ix 44 Normal, 11 37 Normal distribution, 31 Normal period, length of a, 141, 146 Normal station, 141 Normal values, 37, 51 Normals, book of, 51 climatological, 148 Norwegian school, 206 Noyes, G. H., 117, 119 Numerical integration, 82 Numerical series, smoothing of, 68 Observatorio Central de Manila, 7 Oceanity, 195 Organization, International Meteorolog- ical, 14 Orographic microclimatology, 14 Overlapping sums, 69 Partial drought, 124 Peakedness, 41 Pearson, K., 45, 152 Percentiles, 39 Period, auxiliary table for calculating the length of a, 200 normal, 141 if. Periodic phenomena, 170 Periodic variations, 202 Periods of snow cover, 126 Phase angle, 70 Physiological processes and cooling power, 114 Pleions, 178 Point cloud, 157 Pollak, L. W., viii, 17, 19, 44, 72, 74, 80 102, 133, 157, 165, 167 Pollak's tables to harmonic analysis, 73 Polo, Marco, 212 Positive excess, 38 Potential water power, 4 Precipitation, 118 absolute probability of, 53 amounts of, within certain periods, 119 annual course of the amounts of, 120 anomalies of, 180 on an average day of the month, 121 change with height, 202 duration of, 54 effectiveness of, 196 ff. method of measuring, 4, 15 profiles of, 183 variability of, vs. amounts of, 185 Pressure, atmospheric, 100 of water vapor, 100 Primitive characteristics, 17 Primitive elements, 6 Probability, 35 absolute, of precipitation, 53 of a day with precipitation, 118 Probability integral, 35 Probable error, 36 ratio between and correlation coeffi- cient, 152 calculation of (auxiliary table), 218 Projections for isoline maps, 169 Quartiles, 24 Quasi-constancy, 150 of differences, 131 ff. of ratios, 132 ff . Quotients, method of, 142 explanation of the method, 143 Partial correlation, ix, 161, 164 Partial correlation coefficients, 164 Radiation, 212 local influences of, 15 226 METHODS IN CLIMATOLOGY Rain, histogram, 127 intensity, 54, 122 spells, 124 Rain measurements in Palestine, 1 Rainfall, 122 Rain-gauge, diameter of, 16 heights of the rim of, 16 in India, 400 B.C., 1 shielded, 15 Random samples, 53 Range of the extremes, #-years average of, 30 Range of temperature, aperiodical, 88, 89 Range of variation, 17, 19, 25 Rareness of a climatic event, 36 "Ratio of variation," 52 Ratios, quasi-constancy of, 132 Reduction, to a given level (temperatures), 171 of climatological averages to a certain period, 139 of cloudiness to the "true mean," 91 to the "true mean," 81 to a normal period, limits of the method of, 144 of wind velocity to the "true mean," 92 Reed, G., Jr., 211 Reed, W. G., 95 Register, International Climatological, 10, 81 Regression coefficient, 160 Regression equation, ix, 157 ff. Relative amplitudes, 78 Relative frequency, 19 Relative humidity, 4 Relative pluviometric coefficient, 120, 121 Relative shunshine duration and cloudi- ness, 115 Relative temperatures, 98 Relative variability, 51, 122 "Relief energy," 171, 173, 176 Renk, J. J., 201 "Roseau mondial," 2, 5, 7 Resultant run of the wind, 101, 104 Resultant wind velocity, 101, 105 Richey, F. D., 215 Riesbol, H. S., 15 Rietz, H. L., 46, 64 Right skewness, 44 Rotch, A. L. f 15 Run of the wind, 104 Sable, E., 97 Sample of the variate, 18 Scalar quantities, 81 Scatter of a variate, 38 Schell, I. I., 149 Schreier, O., 138 Schuster, Sir Arthur, 80 Season, auxiliary table for calculating the length of a, 220 cold, normal, warm, 207-210 growing, 95, 203 length of dry, 205 length of hot, 205 of snow-cover, 125 "Secular Station," 13, 141 Self-registering instruments, records of, 202 Semi-average method, 61 Sequence of signs, 134 Series, absolutely homogeneous climato- logical, 131 ff. break of, 12 convergence of, 78 discontinuity of, 12 Sextiles, 39 Shaw, Sir Napier, 5, 7, 31, 100, 160, 161, 169, 186, 187 Shelter, thermometer, 15 Shielded rain-gauge, 15 Showalter, A. K., 193 Signs, change of, 134 sequence of, 134 % Silcox, F. A., 215 Skewness, 42, 44 Sky, brightness of the, 57 Smithsonian tables, 101, 105 Smoothing of numerical series, 58, 68 ff. Snow, 124 depth, of freshly fallen, 125 Snow cover, 95 and altitude, 180 duration of a, 125 periods of, 126 season of, 125 Snowfall, days with, 119 duration value of, 126 Special statistical characteristics, 30 Spells, of cold days, 95 of dry and wet days, 124 of hot days, 95 of ice days, 96 of tropical days, 96 of summer days, 96 Spilhaus, A. F., 168 Spitaler, R., 196 Standard curves, 175 Standard deviation, 31, 124 of the arithmetical mean, 33 Standard distribution of an element, 67, 174 INDEX 227 Standard equation, 157 Standard period for climatological nor- mals, 148 Static climatology, 200 Stations of different order, 5 ff. Statistical characteristics, 17 Steadiness of the wind, 106, 189 Steinhauser, F., 25, 126, 201 Steppe, timber forest, and desert, climatic borderlines between, 196 ff. Stevenson's screen, 13 Stone, R. G., 113, 126, 193 Stratification of the air, 14, 15 Streamlines, maps of, 188 " Subnormal ," 37 SOring, R., 177, 178, 179 Summer days, spells of, 96 Sunshine, duration of, vs. altitude, 180 and cloudiness, 55 Supan, A., 196, 197 "Superior force," 36 Superposition of waves, 71 Switzer, I. E., 211 Swoboda, G., 196 Synthesis of climatic elements, 211 Tables to the harmonic analysis, 73 Temperature, absolute, 200 actual, 172 aperiodical range of, 88, 89 of artificial heating, 94, 95 asymmetry of the annual course of, 99 change with height, 202 cumulative, 96 daily course of, 81 distribution of, in mountainous regions, 170 duration of a certain, 94 equivalent, 108 and geographical latitude, 177 of germination, 94 of the human skin, 1 14 inversion, 13, 65, 170 mapping of, in a mountainous region, 171 reduction of, to a given level, 171, 173 relative, 98 surges, 49, 50 threshold of a certain, 92 true, of the air, 13 Temperature-height curve, 174 Tercentesimal, 200 Thermal stratification, 14, 15 Thermomeions, 180 Thermometer, invention of, 1 Thermometer shelter, 15 Thermopleions, 178 Thornthwaite, C. W., 198 Thresholds of temperature, 92 Timber forest and steppe, climatic border- line between, 196 ff. Times of the extremes, 78 Tippett, L. H. C., 34 Trajectories, 188 Tree-growth analysis, viii Trewartha, G. I., 186, 193 Triangulation, climatological, 138 Tropical days, spells of, 96 "True azimuth" of wind direction, 103 "True Mean," correction of the, 81 ff. reduction of wind velocity to the, 92 "True phase angle," 75 True temperature of the air, 13 Turning point of a curve, 66 Unilaterally limited variates, 39 United States Weather Bureau, viii, 7, 9, 11, 15, 16, 17, 84, 89, 97, 115, 121, 125, 201, 216, 217 Upper quartile, 28 Variability, absolute, 51, 122 average, 31, 46 interannual, 50 interdiurnal, 48 interhourly, 50 interseasonal, 50 intersequential, 50 relative, 50, 122 Variability of precipitation vs. amounts of precipitation, 165 Variance, 38, 78 Variate, 17 bilaterally limited, 39 dispersion of a, 38 unilaterally limited, 39 Variations, aperiodic, 202 periodic, 202 range of, 19 ratio of, 52 Varney, B. M., 50 Vectorial quantities, 81 Vegetative period, 4, 95, 174, 176, 203 Visher, S. S., 205 Wagner, A., 57, 201 Wagner, H., 171 Walker, Sir Gilbert, 80 Wallis, B. C., 122 Ward, R. DeCourcy, 46, 199, 200, 214 Warmings, 49 228 METHODS IN CLIMATOLOGY Water vapor pressure, 100 Waves, superposition of, 71 Weather Bureau, see U. S. Weather Bureau Wegener, A., 178, 189 Weighted mean, 85 Weighted overlapping sums, 69 Wet bulb temperature, 110 Wet spells, 123, 124 Wijkander, A., 187, 188 Willett, H. C., 193, 206 Wilson, W. M., 201 Wind, resultant run of the, 101, 104 resultant velocity, 101, 105 steadiness of the, 106, 189 "true azimuth" of the, 103 Wind directions, frequencies of, 103 azimuth of, 103 Wind roses, 186 vs. air mass analysis, 188 for different elements, 187 Wind "star," 186 ff. Wind velocity, reduction of, to the "true mean," 92 resultant, 101, 105 Winkler, M., 126, 180 "World Weather Records," Yule, C. U., 162 Zenker, W., 196